Ordnung
Gruppe
Echte Untergruppen [ 1]
Eigenschaften
Zykel-Graph
1
Z
1
≅
S
1
≅
A
2
{\displaystyle \mathbb {Z} _{1}\cong S_{1}\cong A_{2}}
(triviale Gruppe )
-
abelsch, zyklisch
2
Z
2
≅
S
2
≅
D
1
{\displaystyle \mathbb {Z} _{2}\cong S_{2}\cong D_{1}}
(Gruppe Z2 )
-
abelsch, einfach , zyklisch, kleinste nichttriviale Gruppe
3
Z
3
≅
A
3
{\displaystyle \mathbb {Z} _{3}\cong A_{3}}
-
abelsch, einfach, zyklisch
4
Z
4
≅
D
i
c
1
{\displaystyle \mathbb {Z} _{4}\cong \mathrm {Dic} _{1}}
Z
2
{\displaystyle \mathbb {Z} _{2}}
abelsch, zyklisch
V
4
≅
Z
2
2
≅
D
2
{\displaystyle V_{4}\cong \mathbb {Z} _{2}^{2}\cong D_{2}}
(Kleinsche Vierergruppe )
3
⋅
Z
2
{\displaystyle 3\cdot \mathbb {Z} _{2}}
abelsch, die kleinste nichtzyklische Gruppe
5
Z
5
{\displaystyle \mathbb {Z} _{5}}
-
abelsch, einfach, zyklisch
6
Z
6
≅
Z
2
×
Z
3
{\displaystyle \mathbb {Z} _{6}\cong \mathbb {Z} _{2}\times \mathbb {Z} _{3}}
Z
3
{\displaystyle \mathbb {Z} _{3}}
,
Z
2
{\displaystyle \mathbb {Z} _{2}}
abelsch, zyklisch
S
3
≅
D
3
{\displaystyle S_{3}\cong D_{3}}
(Symmetrische Gruppe )
Z
3
{\displaystyle \mathbb {Z} _{3}}
,
3
⋅
Z
2
{\displaystyle 3\cdot \mathbb {Z} _{2}}
kleinste nichtabelsche Gruppe
7
Z
7
{\displaystyle \mathbb {Z} _{7}}
-
abelsch, einfach, zyklisch
8
Z
8
{\displaystyle \mathbb {Z} _{8}}
Z
4
{\displaystyle \mathbb {Z} _{4}}
,
Z
2
{\displaystyle \mathbb {Z} _{2}}
abelsch, zyklisch
Z
2
×
Z
4
{\displaystyle \mathbb {Z} _{2}\times \mathbb {Z} _{4}}
2
⋅
Z
4
{\displaystyle 2\cdot \mathbb {Z} _{4}}
,
3
⋅
Z
2
{\displaystyle 3\cdot \mathbb {Z} _{2}}
,
D
2
{\displaystyle D_{2}}
abelsch
Z
2
3
≅
D
2
×
Z
2
{\displaystyle \mathbb {Z} _{2}^{3}\cong D_{2}\times \mathbb {Z} _{2}}
7
⋅
Z
2
{\displaystyle 7\cdot \mathbb {Z} _{2}}
,
7
⋅
D
2
{\displaystyle 7\cdot D_{2}}
abelsch
D
4
{\displaystyle D_{4}}
Z
4
{\displaystyle \mathbb {Z} _{4}}
,
2
⋅
D
2
{\displaystyle 2\cdot D_{2}}
,
5
⋅
Z
2
{\displaystyle 5\cdot \mathbb {Z} _{2}}
nichtabelsch
Q
8
≅
D
i
c
2
{\displaystyle Q_{8}\cong \mathrm {Dic} _{2}}
(Quaternionengruppe )
3
⋅
Z
4
{\displaystyle 3\cdot \mathbb {Z} _{4}}
,
Z
2
{\displaystyle \mathbb {Z} _{2}}
nichtabelsch; die kleinste hamiltonsche Gruppe
9
Z
9
{\displaystyle \mathbb {Z} _{9}}
Z
3
{\displaystyle \mathbb {Z} _{3}}
abelsch, zyklisch
Z
3
2
{\displaystyle \mathbb {Z} _{3}^{2}}
4
⋅
Z
3
{\displaystyle 4\cdot \mathbb {Z} _{3}}
abelsch
10
Z
10
≅
Z
2
×
Z
5
{\displaystyle \mathbb {Z} _{10}\cong \mathbb {Z} _{2}\times \mathbb {Z} _{5}}
Z
5
{\displaystyle \mathbb {Z} _{5}}
,
Z
2
{\displaystyle \mathbb {Z} _{2}}
abelsch, zyklisch
D
5
{\displaystyle D_{5}}
Z
5
{\displaystyle \mathbb {Z} _{5}}
,
5
⋅
Z
2
{\displaystyle 5\cdot \mathbb {Z} _{2}}
nichtabelsch
11
Z
11
{\displaystyle \mathbb {Z} _{11}}
-
abelsch, einfach, zyklisch
12
Z
12
≅
Z
4
×
Z
3
{\displaystyle \mathbb {Z} _{12}\cong \mathbb {Z} _{4}\times \mathbb {Z} _{3}}
Z
6
{\displaystyle \mathbb {Z} _{6}}
,
Z
4
{\displaystyle \mathbb {Z} _{4}}
,
Z
3
{\displaystyle \mathbb {Z} _{3}}
,
Z
2
{\displaystyle \mathbb {Z} _{2}}
abelsch, zyklisch
Z
2
×
Z
6
≅
Z
2
2
×
Z
3
≅
D
2
×
Z
3
{\displaystyle \mathbb {Z} _{2}\times \mathbb {Z} _{6}\cong \mathbb {Z} _{2}^{2}\times \mathbb {Z} _{3}\cong D_{2}\times \mathbb {Z} _{3}}
3
⋅
Z
6
{\displaystyle 3\cdot \mathbb {Z} _{6}}
,
Z
3
{\displaystyle \mathbb {Z} _{3}}
,
D
2
{\displaystyle D_{2}}
,
3
⋅
Z
2
{\displaystyle 3\cdot \mathbb {Z} _{2}}
abelsch
D
6
≅
D
3
×
Z
2
{\displaystyle D_{6}\cong D_{3}\times \mathbb {Z} _{2}}
Z
6
{\displaystyle \mathbb {Z} _{6}}
,
2
⋅
D
3
{\displaystyle 2\cdot D_{3}}
,
3
⋅
D
2
{\displaystyle 3\cdot D_{2}}
,
Z
3
{\displaystyle \mathbb {Z} _{3}}
,
7
⋅
Z
2
{\displaystyle 7\cdot \mathbb {Z} _{2}}
nichtabelsch
A
4
{\displaystyle A_{4}}
(Gruppe A4 )
D
2
{\displaystyle D_{2}}
,
4
⋅
Z
3
{\displaystyle 4\cdot \mathbb {Z} _{3}}
,
3
⋅
Z
2
{\displaystyle 3\cdot \mathbb {Z} _{2}}
nichtabelsch; kleinste Gruppe, die zeigt, dass die Umkehrung des Satzes von Lagrange nicht stimmt: keine Untergruppe der Ordnung 6
D
i
c
3
{\displaystyle \mathrm {Dic} _{3}}
(hier Verknüpfungstafel )
Z
6
{\displaystyle \mathbb {Z} _{6}}
,
3
⋅
Z
4
{\displaystyle 3\cdot \mathbb {Z} _{4}}
,
Z
3
{\displaystyle \mathbb {Z} _{3}}
,
Z
2
{\displaystyle \mathbb {Z} _{2}}
nichtabelsch
13
Z
13
{\displaystyle \mathbb {Z} _{13}}
-
abelsch, einfach, zyklisch
14
Z
14
≅
Z
2
×
Z
7
{\displaystyle \mathbb {Z} _{14}\cong \mathbb {Z} _{2}\times \mathbb {Z} _{7}}
Z
7
{\displaystyle \mathbb {Z} _{7}}
,
Z
2
{\displaystyle \mathbb {Z} _{2}}
abelsch, zyklisch
D
7
{\displaystyle D_{7}}
Z
7
{\displaystyle \mathbb {Z} _{7}}
,
7
⋅
Z
2
{\displaystyle 7\cdot \mathbb {Z} _{2}}
nichtabelsch
15
Z
15
≅
Z
3
×
Z
5
{\displaystyle \mathbb {Z} _{15}\cong \mathbb {Z} _{3}\times \mathbb {Z} _{5}}
Z
5
{\displaystyle \mathbb {Z} _{5}}
,
Z
3
{\displaystyle \mathbb {Z} _{3}}
abelsch, zyklisch (siehe „Jede Gruppe der Ordnung 15 ist zyklisch.“ )
16
Z
16
{\displaystyle \mathbb {Z} _{16}}
Z
8
{\displaystyle \mathbb {Z} _{8}}
,
Z
4
{\displaystyle \mathbb {Z} _{4}}
,
Z
2
{\displaystyle \mathbb {Z} _{2}}
abelsch, zyklisch
Z
2
4
{\displaystyle \mathbb {Z} _{2}^{4}}
15
⋅
Z
2
{\displaystyle 15\cdot \mathbb {Z} _{2}}
,
35
⋅
D
2
{\displaystyle 35\cdot D_{2}}
,
15
⋅
Z
2
3
{\displaystyle 15\cdot \mathbb {Z} _{2}^{3}}
abelsch
Z
4
×
Z
2
2
{\displaystyle \mathbb {Z} _{4}\times \mathbb {Z} _{2}^{2}}
7
⋅
Z
2
{\displaystyle 7\cdot \mathbb {Z} _{2}}
,
4
⋅
Z
4
{\displaystyle 4\cdot \mathbb {Z} _{4}}
,
7
⋅
D
2
{\displaystyle 7\cdot D_{2}}
,
Z
2
3
{\displaystyle \mathbb {Z} _{2}^{3}}
,
6
⋅
Z
4
×
Z
2
{\displaystyle 6\cdot \mathbb {Z} _{4}\times \mathbb {Z} _{2}}
abelsch
Z
8
×
Z
2
{\displaystyle \mathbb {Z} _{8}\times \mathbb {Z} _{2}}
3
⋅
Z
2
{\displaystyle 3\cdot \mathbb {Z} _{2}}
,
2
⋅
Z
4
{\displaystyle 2\cdot \mathbb {Z} _{4}}
,
D
2
{\displaystyle D_{2}}
,
2
⋅
Z
8
{\displaystyle 2\cdot \mathbb {Z} _{8}}
,
Z
4
×
Z
2
{\displaystyle \mathbb {Z} _{4}\times \mathbb {Z} _{2}}
abelsch
Z
4
2
{\displaystyle \mathbb {Z} _{4}^{2}}
3
⋅
Z
2
{\displaystyle 3\cdot \mathbb {Z} _{2}}
,
6
⋅
Z
4
{\displaystyle 6\cdot \mathbb {Z} _{4}}
,
D
2
{\displaystyle D_{2}}
,
3
⋅
Z
4
×
Z
2
{\displaystyle 3\cdot \mathbb {Z} _{4}\times \mathbb {Z} _{2}}
abelsch
D
8
{\displaystyle D_{8}}
Z
8
{\displaystyle \mathbb {Z} _{8}}
,
2
⋅
D
4
{\displaystyle 2\cdot D_{4}}
,
4
⋅
D
2
{\displaystyle 4\cdot D_{2}}
,
Z
4
{\displaystyle \mathbb {Z} _{4}}
,
9
⋅
Z
2
{\displaystyle 9\cdot \mathbb {Z} _{2}}
nichtabelsch
D
4
×
Z
2
{\displaystyle D_{4}\times \mathbb {Z} _{2}}
4
⋅
D
4
{\displaystyle 4\cdot D_{4}}
,
Z
4
×
Z
2
{\displaystyle \mathbb {Z} _{4}\times \mathbb {Z} _{2}}
,
2
⋅
Z
2
3
{\displaystyle 2\cdot \mathbb {Z} _{2}^{3}}
,
13
⋅
Z
2
2
{\displaystyle 13\cdot \mathbb {Z} _{2}^{2}}
,
2
⋅
Z
4
{\displaystyle 2\cdot \mathbb {Z} _{4}}
,
11
⋅
Z
2
{\displaystyle 11\cdot \mathbb {Z} _{2}}
nichtabelsch
Q
16
≅
D
i
c
4
{\displaystyle Q_{16}\cong \mathrm {Dic_{4}} }
Z
8
{\displaystyle \mathbb {Z} _{8}}
,
2
⋅
Q
8
{\displaystyle 2\cdot Q_{8}}
,
5
⋅
Z
4
{\displaystyle 5\cdot \mathbb {Z} _{4}}
,
Z
2
{\displaystyle \mathbb {Z} _{2}}
nichtabelsch
Q
8
×
Z
2
{\displaystyle Q_{8}\times \mathbb {Z} _{2}}
3
⋅
Z
2
×
Z
4
{\displaystyle 3\cdot \mathbb {Z} _{2}\times \mathbb {Z} _{4}}
,
4
⋅
Q
8
{\displaystyle 4\cdot Q_{8}}
,
6
⋅
Z
4
{\displaystyle 6\cdot \mathbb {Z} _{4}}
,
Z
2
×
Z
2
{\displaystyle \mathbb {Z} _{2}\times \mathbb {Z} _{2}}
,
3
⋅
Z
2
{\displaystyle 3\cdot \mathbb {Z} _{2}}
nichtabelsch, hamiltonsche Gruppe
Quasi-Diedergruppe
Z
8
{\displaystyle \mathbb {Z} _{8}}
,
Q
8
{\displaystyle Q_{8}}
,
D
4
{\displaystyle D_{4}}
,
3
⋅
Z
4
{\displaystyle 3\cdot \mathbb {Z} _{4}}
,
2
⋅
Z
2
×
Z
2
{\displaystyle 2\cdot \mathbb {Z} _{2}\times \mathbb {Z} _{2}}
,
5
⋅
Z
2
{\displaystyle 5\cdot \mathbb {Z} _{2}}
nichtabelsch
Nichtabelsche nicht-hamiltonsche modulare Gruppe
2
⋅
Z
8
{\displaystyle 2\cdot \mathbb {Z} _{8}}
,
Z
4
×
Z
2
{\displaystyle \mathbb {Z} _{4}\times \mathbb {Z} _{2}}
,
2
⋅
Z
4
{\displaystyle 2\cdot \mathbb {Z} _{4}}
,
Z
2
×
Z
2
{\displaystyle \mathbb {Z} _{2}\times \mathbb {Z} _{2}}
,
3
⋅
Z
2
{\displaystyle 3\cdot \mathbb {Z} _{2}}
nichtabelsch
Semidirektes Produkt
Z
4
⋊
Z
4
{\displaystyle \mathbb {Z} _{4}\rtimes \mathbb {Z} _{4}}
(siehe hier )
3
⋅
Z
2
×
Z
4
{\displaystyle 3\cdot \mathbb {Z} _{2}\times \mathbb {Z} _{4}}
,
6
⋅
Z
4
{\displaystyle 6\cdot \mathbb {Z} _{4}}
,
Z
2
×
Z
2
{\displaystyle \mathbb {Z} _{2}\times \mathbb {Z} _{2}}
,
3
⋅
Z
2
{\displaystyle 3\cdot \mathbb {Z} _{2}}
nichtabelsch
Die durch Pauli-Matrizen erzeugte Gruppe.
3
⋅
Z
2
×
Z
4
{\displaystyle 3\cdot \mathbb {Z} _{2}\times \mathbb {Z} _{4}}
,
3
⋅
D
4
{\displaystyle 3\cdot D_{4}}
,
Q
8
{\displaystyle Q_{8}}
,
4
⋅
Z
4
{\displaystyle 4\cdot \mathbb {Z} _{4}}
,
3
⋅
Z
2
×
Z
2
{\displaystyle 3\cdot \mathbb {Z} _{2}\times \mathbb {Z} _{2}}
,
7
⋅
Z
2
{\displaystyle 7\cdot \mathbb {Z} _{2}}
nichtabelsch
G
4
,
4
=
V
4
⋊
Z
4
{\displaystyle G_{4,4}=V_{4}\rtimes \mathbb {Z} _{4}}
2
⋅
Z
2
×
Z
4
{\displaystyle 2\cdot \mathbb {Z} _{2}\times \mathbb {Z} _{4}}
,
Z
2
×
Z
2
×
Z
2
{\displaystyle \mathbb {Z} _{2}\times \mathbb {Z} _{2}\times \mathbb {Z} _{2}}
,
4
⋅
Z
4
{\displaystyle 4\cdot \mathbb {Z} _{4}}
,
7
⋅
Z
2
×
Z
2
{\displaystyle 7\cdot \mathbb {Z} _{2}\times \mathbb {Z} _{2}}
,
7
⋅
Z
2
{\displaystyle 7\cdot \mathbb {Z} _{2}}
nichtabelsch
17
Z
17
{\displaystyle \mathbb {Z} _{17}}
-
abelsch, einfach, zyklisch
18
Z
18
≅
Z
9
×
Z
2
{\displaystyle \mathbb {Z} _{18}\cong \mathbb {Z} _{9}\times \mathbb {Z} _{2}}
Z
9
,
{\displaystyle \mathbb {Z} _{9},}
Z
6
,
{\displaystyle \mathbb {Z} _{6},}
Z
3
,
{\displaystyle \mathbb {Z} _{3},}
Z
2
{\displaystyle \mathbb {Z} _{2}}
abelsch, zyklisch
Z
6
×
Z
3
{\displaystyle \mathbb {Z} _{6}\times \mathbb {Z} _{3}}
Z
3
2
,
{\displaystyle \mathbb {Z} _{3}^{2},}
4
⋅
Z
6
,
{\displaystyle 4\cdot \mathbb {Z} _{6},}
4
⋅
Z
3
,
{\displaystyle 4\cdot \mathbb {Z} _{3},}
Z
2
{\displaystyle \mathbb {Z} _{2}}
abelsch
D
9
{\displaystyle D_{9}}
Z
9
,
{\displaystyle \mathbb {Z} _{9},}
3
⋅
D
3
,
{\displaystyle 3\cdot D_{3},}
Z
3
,
{\displaystyle \mathbb {Z} _{3},}
9
⋅
Z
2
{\displaystyle 9\cdot \mathbb {Z} _{2}}
nichtabelsch
S
3
×
Z
3
{\displaystyle S_{3}\times \mathbb {Z} _{3}}
Z
3
2
,
{\displaystyle \mathbb {Z} _{3}^{2},}
D
3
,
{\displaystyle D_{3},}
3
⋅
Z
6
,
{\displaystyle 3\cdot \mathbb {Z} _{6},}
4
⋅
Z
3
,
{\displaystyle 4\cdot \mathbb {Z} _{3},}
3
⋅
Z
2
{\displaystyle 3\cdot \mathbb {Z} _{2}}
nichtabelsch
(
Z
3
×
Z
3
)
⋊
α
Z
2
{\displaystyle (\mathbb {Z} _{3}\times \mathbb {Z} _{3})\rtimes _{\alpha }\mathbb {Z} _{2}}
mit
α
(
1
)
=
(
2
0
0
2
)
{\displaystyle \alpha (1)={\begin{pmatrix}2&0\\0&2\end{pmatrix}}}
Z
3
2
,
{\displaystyle \mathbb {Z} _{3}^{2},}
12
⋅
D
3
,
{\displaystyle 12\cdot D_{3},}
4
⋅
Z
3
,
{\displaystyle 4\cdot \mathbb {Z} _{3},}
9
⋅
Z
2
{\displaystyle 9\cdot \mathbb {Z} _{2}}
nichtabelsch
19
Z
19
{\displaystyle \mathbb {Z} _{19}}
-
abelsch, einfach, zyklisch
20
Z
20
≅
Z
5
×
Z
4
{\displaystyle \mathbb {Z} _{20}\cong \mathbb {Z} _{5}\times \mathbb {Z} _{4}}
Z
10
,
{\displaystyle \mathbb {Z} _{10},}
Z
5
,
{\displaystyle \mathbb {Z} _{5},}
Z
4
,
{\displaystyle \mathbb {Z} _{4},}
Z
2
{\displaystyle \mathbb {Z} _{2}}
abelsch, zyklisch
Z
10
×
Z
2
≅
Z
5
×
Z
2
×
Z
2
{\displaystyle \mathbb {Z} _{10}\times \mathbb {Z} _{2}\cong \mathbb {Z} _{5}\times \mathbb {Z} _{2}\times \mathbb {Z} _{2}}
3
⋅
Z
10
,
{\displaystyle 3\cdot \mathbb {Z} _{10},}
Z
5
,
{\displaystyle \mathbb {Z} _{5},}
D
2
,
{\displaystyle D_{2},}
3
⋅
Z
2
{\displaystyle 3\cdot \mathbb {Z} _{2}}
abelsch
Q
20
≅
D
i
c
5
{\displaystyle Q_{20}\cong \mathrm {Dic} _{5}}
Z
10
,
{\displaystyle \mathbb {Z} _{10},}
Z
5
,
{\displaystyle \mathbb {Z} _{5},}
5
⋅
Z
4
,
{\displaystyle 5\cdot \mathbb {Z} _{4},}
Z
2
{\displaystyle \mathbb {Z} _{2}}
nichtabelsch
Z
5
⋊
Z
4
≅
{\displaystyle \mathbb {Z} _{5}\rtimes \mathbb {Z} _{4}\cong }
AGL1 (5)
D
5
,
{\displaystyle D_{5},}
Z
5
,
{\displaystyle \mathbb {Z} _{5},}
5
⋅
Z
4
,
{\displaystyle 5\cdot \mathbb {Z} _{4},}
5
⋅
Z
2
{\displaystyle 5\cdot \mathbb {Z} _{2}}
nichtabelsch
D
10
≅
D
5
×
Z
2
{\displaystyle D_{10}\cong D_{5}\times \mathbb {Z} _{2}}
Z
10
,
{\displaystyle \mathbb {Z} _{10},}
D
5
,
{\displaystyle D_{5},}
Z
5
,
{\displaystyle \mathbb {Z} _{5},}
5
⋅
V
4
,
{\displaystyle 5\cdot V_{4},}
11
⋅
Z
2
{\displaystyle 11\cdot \mathbb {Z} _{2}}
nichtabelsch
21
Z
21
≅
Z
7
×
Z
3
{\displaystyle \mathbb {Z} _{21}\cong \mathbb {Z} _{7}\times \mathbb {Z} _{3}}
Z
7
,
{\displaystyle \mathbb {Z} _{7},}
Z
3
{\displaystyle \mathbb {Z} _{3}}
abelsch, zyklisch
Z
7
⋊
Z
3
{\displaystyle \mathbb {Z} _{7}\rtimes \mathbb {Z} _{3}}
Z
7
,
{\displaystyle \mathbb {Z} _{7},}
7
⋅
Z
3
{\displaystyle 7\cdot \mathbb {Z} _{3}}
nichtabelsch
22
Z
22
≅
Z
11
×
Z
2
{\displaystyle \mathbb {Z} _{22}\cong \mathbb {Z} _{11}\times \mathbb {Z} _{2}}
Z
11
,
{\displaystyle \mathbb {Z} _{11},}
Z
2
{\displaystyle \mathbb {Z} _{2}}
abelsch, zyklisch
D
11
{\displaystyle D_{11}}
Z
11
,
{\displaystyle \mathbb {Z} _{11},}
11
⋅
Z
2
{\displaystyle 11\cdot \mathbb {Z} _{2}}
nichtabelsch
23
Z
23
{\displaystyle \mathbb {Z} _{23}}
-
abelsch, einfach, zyklisch
24
Z
24
≅
Z
8
×
Z
3
{\displaystyle \mathbb {Z} _{24}\cong \mathbb {Z} _{8}\times \mathbb {Z} _{3}}
Z
12
,
{\displaystyle \mathbb {Z} _{12},}
Z
8
,
{\displaystyle \mathbb {Z} _{8},}
Z
6
,
{\displaystyle \mathbb {Z} _{6},}
Z
4
,
{\displaystyle \mathbb {Z} _{4},}
Z
3
,
{\displaystyle \mathbb {Z} _{3},}
Z
2
{\displaystyle \mathbb {Z} _{2}}
abelsch, zyklisch
Z
12
×
Z
2
≅
Z
6
×
Z
4
≅
{\displaystyle \mathbb {Z} _{12}\times \mathbb {Z} _{2}\cong \mathbb {Z} _{6}\times \mathbb {Z} _{4}\cong }
Z
4
×
Z
3
×
Z
2
{\displaystyle \mathbb {Z} _{4}\times \mathbb {Z} _{3}\times \mathbb {Z} _{2}}
Z
12
,
{\displaystyle \mathbb {Z} _{12},}
Z
6
,
{\displaystyle \mathbb {Z} _{6},}
Z
4
,
{\displaystyle \mathbb {Z} _{4},}
Z
3
,
{\displaystyle \mathbb {Z} _{3},}
Z
2
{\displaystyle \mathbb {Z} _{2}}
abelsch
Z
6
×
D
2
≅
Z
3
×
Z
2
3
{\displaystyle \mathbb {Z} _{6}\times D_{2}\cong \mathbb {Z} _{3}\times \mathbb {Z} _{2}^{3}}
Z
6
,
{\displaystyle \mathbb {Z} _{6},}
Z
3
,
{\displaystyle \mathbb {Z} _{3},}
Z
2
{\displaystyle \mathbb {Z} _{2}}
abelsch
Z
3
⋊
Z
8
{\displaystyle \mathbb {Z} _{3}\rtimes \mathbb {Z} _{8}}
Z
12
,
{\displaystyle \mathbb {Z} _{12},}
3
⋅
Z
8
,
{\displaystyle 3\cdot \mathbb {Z} _{8},}
Z
6
,
{\displaystyle \mathbb {Z} _{6},}
Z
4
,
{\displaystyle \mathbb {Z} _{4},}
Z
3
,
{\displaystyle \mathbb {Z} _{3},}
Z
2
{\displaystyle \mathbb {Z} _{2}}
nichtabelsch
SL (2,3)
≅
Q
8
⋊
Z
3
{\displaystyle \cong Q_{8}\rtimes \mathbb {Z} _{3}}
Q
8
,
{\displaystyle Q_{8},}
4
⋅
Z
6
,
{\displaystyle 4\cdot \mathbb {Z} _{6},}
3
⋅
Z
4
,
{\displaystyle 3\cdot \mathbb {Z} _{4},}
4
⋅
Z
3
,
{\displaystyle 4\cdot \mathbb {Z} _{3},}
Z
2
{\displaystyle \mathbb {Z} _{2}}
nichtabelsch
Q
24
≅
Z
3
×
Q
8
{\displaystyle Q_{24}\cong \mathbb {Z} _{3}\times Q_{8}}
Z
12
,
{\displaystyle \mathbb {Z} _{12},}
2
⋅
Q
12
,
{\displaystyle 2\cdot Q_{12},}
3
⋅
Q
8
,
{\displaystyle 3\cdot Q_{8},}
Z
6
,
{\displaystyle \mathbb {Z} _{6},}
7
⋅
Z
4
,
{\displaystyle 7\cdot \mathbb {Z} _{4},}
Z
3
,
{\displaystyle \mathbb {Z} _{3},}
Z
2
{\displaystyle \mathbb {Z} _{2}}
nichtabelsch
D
3
×
Z
4
≅
S
3
×
Z
4
{\displaystyle D_{3}\times \mathbb {Z} _{4}\cong S_{3}\times \mathbb {Z} _{4}}
Z
12
,
{\displaystyle \mathbb {Z} _{12},}
Q
12
,
{\displaystyle Q_{12},}
D
6
,
{\displaystyle D_{6},}
3
⋅
Z
4
×
Z
2
,
{\displaystyle 3\cdot \mathbb {Z} _{4}\times \mathbb {Z} _{2},}
Z
6
,
{\displaystyle \mathbb {Z} _{6},}
2
⋅
D
3
,
{\displaystyle 2\cdot D_{3},}
4
⋅
Z
4
,
{\displaystyle 4\cdot \mathbb {Z} _{4},}
3
⋅
D
2
,
{\displaystyle 3\cdot D_{2},}
Z
3
,
{\displaystyle \mathbb {Z} _{3},}
7
⋅
Z
2
{\displaystyle 7\cdot \mathbb {Z} _{2}}
nichtabelsch
D
12
{\displaystyle D_{12}}
Z
12
,
{\displaystyle \mathbb {Z} _{12},}
2
⋅
D
6
,
{\displaystyle 2\cdot D_{6},}
3
⋅
D
4
,
{\displaystyle 3\cdot D_{4},}
Z
6
,
{\displaystyle \mathbb {Z} _{6},}
4
⋅
D
3
,
{\displaystyle 4\cdot D_{3},}
Z
4
,
{\displaystyle \mathbb {Z} _{4},}
6
⋅
D
2
,
{\displaystyle 6\cdot D_{2},}
Z
3
,
{\displaystyle \mathbb {Z} _{3},}
13
⋅
Z
2
{\displaystyle 13\cdot \mathbb {Z} _{2}}
nichtabelsch
Q
12
×
Z
2
≅
(
Z
3
⋊
Z
4
)
×
Z
2
{\displaystyle Q_{12}\times \mathbb {Z} _{2}\cong (\mathbb {Z} _{3}\rtimes \mathbb {Z} _{4})\times \mathbb {Z} _{2}}
Z
6
×
Z
2
,
{\displaystyle \mathbb {Z} _{6}\times \mathbb {Z} _{2},}
2
⋅
Q
12
,
{\displaystyle 2\cdot Q_{12},}
3
⋅
Z
4
×
Z
2
,
{\displaystyle 3\cdot \mathbb {Z} _{4}\times \mathbb {Z} _{2},}
3
⋅
Z
6
,
{\displaystyle 3\cdot \mathbb {Z} _{6},}
6
⋅
Z
4
,
{\displaystyle 6\cdot \mathbb {Z} _{4},}
D
2
,
{\displaystyle D_{2},}
Z
3
,
{\displaystyle \mathbb {Z} _{3},}
3
⋅
Z
2
{\displaystyle 3\cdot \mathbb {Z} _{2}}
nichtabelsch
(
Z
6
×
Z
2
)
⋊
Z
2
≅
Z
3
⋊
D
4
{\displaystyle (\mathbb {Z} _{6}\times \mathbb {Z} _{2})\rtimes \mathbb {Z} _{2}\cong \mathbb {Z} _{3}\rtimes D_{4}}
Z
6
×
Z
2
,
{\displaystyle \mathbb {Z} _{6}\times \mathbb {Z} _{2},}
Q
12
,
{\displaystyle Q_{12},}
D
3
,
{\displaystyle D_{3},}
3
⋅
D
4
,
{\displaystyle 3\cdot D_{4},}
3
⋅
Z
6
,
{\displaystyle 3\cdot \mathbb {Z} _{6},}
2
⋅
D
3
,
{\displaystyle 2\cdot D_{3},}
3
⋅
Z
4
,
{\displaystyle 3\cdot \mathbb {Z} _{4},}
4
⋅
D
2
,
{\displaystyle 4\cdot D_{2},}
Z
3
,
{\displaystyle \mathbb {Z} _{3},}
9
⋅
Z
2
{\displaystyle 9\cdot \mathbb {Z} _{2}}
nichtabelsch
D
4
×
Z
3
{\displaystyle D_{4}\times \mathbb {Z} _{3}}
Z
12
,
{\displaystyle \mathbb {Z} _{12},}
2
⋅
Z
6
×
Z
2
,
{\displaystyle 2\cdot \mathbb {Z} _{6}\times \mathbb {Z} _{2},}
D
4
,
{\displaystyle D_{4},}
5
⋅
Z
6
,
{\displaystyle 5\cdot \mathbb {Z} _{6},}
Z
4
,
{\displaystyle \mathbb {Z} _{4},}
2
⋅
D
2
,
{\displaystyle 2\cdot D_{2},}
Z
3
,
{\displaystyle \mathbb {Z} _{3},}
5
⋅
Z
2
{\displaystyle 5\cdot \mathbb {Z} _{2}}
nichtabelsch
Q
8
×
Z
3
{\displaystyle Q_{8}\times \mathbb {Z} _{3}}
3
⋅
Z
12
,
{\displaystyle 3\cdot \mathbb {Z} _{12},}
Q
8
,
{\displaystyle Q_{8},}
Z
6
,
{\displaystyle \mathbb {Z} _{6},}
3
⋅
Z
4
,
{\displaystyle 3\cdot \mathbb {Z} _{4},}
Z
3
,
{\displaystyle \mathbb {Z} _{3},}
Z
2
{\displaystyle \mathbb {Z} _{2}}
nichtabelsch
S
4
{\displaystyle S_{4}}
A
4
,
{\displaystyle A_{4},}
3
⋅
D
4
,
{\displaystyle 3\cdot D_{4},}
4
⋅
D
3
,
{\displaystyle 4\cdot D_{3},}
3
⋅
Z
4
,
{\displaystyle 3\cdot \mathbb {Z} _{4},}
4
⋅
D
2
,
{\displaystyle 4\cdot D_{2},}
4
⋅
Z
3
,
{\displaystyle 4\cdot \mathbb {Z} _{3},}
9
⋅
Z
2
{\displaystyle 9\cdot \mathbb {Z} _{2}}
nichtabelsch
A
4
×
Z
2
{\displaystyle A_{4}\times \mathbb {Z} _{2}}
A
4
,
{\displaystyle A_{4},}
Z
2
3
,
{\displaystyle \mathbb {Z} _{2}^{3},}
4
⋅
Z
6
,
{\displaystyle 4\cdot \mathbb {Z} _{6},}
7
⋅
D
2
,
{\displaystyle 7\cdot D_{2},}
4
⋅
Z
3
,
{\displaystyle 4\cdot \mathbb {Z} _{3},}
7
⋅
Z
2
{\displaystyle 7\cdot \mathbb {Z} _{2}}
nichtabelsch
D
6
×
Z
2
{\displaystyle D_{6}\times \mathbb {Z} _{2}}
Z
6
×
Z
2
,
{\displaystyle \mathbb {Z} _{6}\times \mathbb {Z} _{2},}
6
⋅
D
6
,
{\displaystyle 6\cdot D_{6},}
3
⋅
Z
2
3
,
{\displaystyle 3\cdot \mathbb {Z} _{2}^{3},}
3
⋅
Z
6
,
{\displaystyle 3\cdot \mathbb {Z} _{6},}
4
⋅
D
3
,
{\displaystyle 4\cdot D_{3},}
19
⋅
D
2
,
{\displaystyle 19\cdot D_{2},}
Z
3
,
{\displaystyle \mathbb {Z} _{3},}
15
⋅
Z
2
{\displaystyle 15\cdot \mathbb {Z} _{2}}
nichtabelsch