Finite Group Databasebeta

← C2×Dic3×C3⋊(C3⋊D3) #13049673

Subgroups — C2×Dic3×C3⋊(C3⋊D3) #13049673

Order 1296.

2136 conjugacy classes of subgroups; only the 394 normal subgroups are shown.

normal subgroups only

This group has 2136 conjugacy classes of subgroups — too many to draw. Only its 394 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C3⋊(C3⋊D3)C211yesmaximal, characteristic
2C2×C3⋊(C3⋊(C3⋊Dic3))C211yesmaximal, characteristic, complemented
3C2×C33×Dic3C211yesmaximal, characteristic, complemented
4Dic3×C3⋊(C3⋊D3)C214yesmaximal, direct product factor
5C2×C4×C3⋊(C3⋊D3)—31nomaximal
6C2×Dic3×C3⋊D3—313nomaximal
7C22×C34C2211yescharacteristic
8C2×C3×C3⋊(C3⋊D3)C414yescomplemented
9C2×C3×C3⋊(C3⋊D3)C2212yes—
10C3⋊(C3⋊(C3⋊Dic3))C2212yescomplemented
11C33×Dic3C2212yescomplemented
12C22×C3⋊(C3⋊D3)D311yescharacteristic
13C22×C3×C3⋊D3—313no—
14C2×C33×C4—31no—
15C2×C3⋊(C3⋊Dic3)—31no—
16C2×C3⋊(C3⋊Dic3)—313no—
17C2×C3⋊(C3⋊Dic3)—613no—
18C4×C3⋊(C3⋊D3)—34no—
19C2×C32×Dic3D3113yescomplemented
20Dic3×C3⋊D3—352no—
21C2×C34C2×C412yescomplemented
22C2×C34C2311yescharacteristic
23C3×C3⋊(C3⋊D3)C2×C414yescomplemented
24C2×C4×C3⋊D3—913no—
25C2×D3×Dic3—913no—
26C22×C33—213no—
27C22×C33C2×D3113yes—
28C22×C33C2×D311yescharacteristic
29C2×C3⋊(C3⋊D3)Dic314yesdirect product factor
30C2×C3⋊(C3⋊D3)C2×D312yes—
31C2×C3×C3⋊D3—352no—
32C2×C3×C3⋊D3—326no—
33C33×C4—32no—
34C3⋊(C3⋊Dic3)—32no—
35C3⋊(C3⋊Dic3)—326no—
36C3⋊(C3⋊Dic3)—626no—
37C32×Dic3C2×D3126yescomplemented
38C34C22×C411yescharacteristic, complemented, Sylow
39C22×C3⋊D3—313no—
40C22×C3×D3—913no—
41C2×C32×C4—313no—
42C2×C3⋊Dic3—913no—
43C2×C3⋊Dic3—913no—
44C2×C3⋊Dic3—1852no—
45C4×C3⋊D3—952no—
46C2×C3×Dic3C3⋊D3113yescomplemented
47D3×Dic3—952no—
48C2×C33—226no—
49C2×C33—213no—
50C2×C33C4×D3126yescomplemented
51C2×C33C2×Dic312yescomplemented
52C2×C33C22×D3113yes—
53C2×C33C22×D311yescharacteristic
54C3⋊(C3⋊D3)C2×Dic314yesdirect product factor
55C3×C3⋊D3—352no—
56C22×Dic3—271no—
57C2×C4×D3—2713no—
58C22×C32—252no—
59C22×C32D32113yes—
60C22×C32C2×C3⋊D3113yes—
61C2×C3⋊D3—352no—
62C2×C3⋊D3—326no—
63C2×C3×D3—952no—
64C2×C3×D3—926no—
65C32×C4—326no—
66C3⋊Dic3—926no—
67C3⋊Dic3—926no—
68C3⋊Dic3—18104no—
69C3×Dic3C2×C3⋊D3126yescomplemented
70C33—213no—
71C33C2×C4×D3113yescomplemented
72C33C22×Dic311yescharacteristic, complemented
73C23×C3—271no—
74C22×D3—913no—
75C2×C3×C4—313no—
76C2×Dic3—2713no—
77C2×Dic3—271no—
78C2×Dic3—274no—
79C2×Dic3—5413no—
80C2×Dic3C3⋊(C3⋊D3)11yescharacteristic, direct product factor
81C4×D3—2752no—
82C2×C32—2104no—
83C2×C32—252no—
84C2×C32D3×Dic3126yescomplemented
85C2×C32C4×C3⋊D3126yescomplemented
86C2×C32C2×D32113yes—
87C2×C32C22×C3⋊D3113yes—
88C3⋊D3—352no—
89C3×D3—952no—
90C22×C4—811noSylow
91C22×C3—213no—
92C22×C3—274no—
93C22×C3—272no—
94C22×C3D3×C3⋊D3113yes—
95C22×C3C2×C3⋊(C3⋊D3)11yescharacteristic
96C2×D3—952no—
97C2×D3—926no—
98C3×C4—326no—
99Dic3—2726no—
100Dic3—272no—
101Dic3—5426no—
102Dic3C2×C3⋊(C3⋊D3)12yesdirect product factor
103C32—252no—
104C32C2×D3×Dic3113yescomplemented
105C32C2×C4×C3⋊D3113yescomplemented
106C23—271no—
107C2×C4—31no—
108C2×C4—814no—
109C2×C4—811no—
110C2×C3—213no—
111C2×C3—226no—
112C2×C3—274no—
113C2×C3Dic3×C3⋊D3126yescomplemented
114C2×C3C4×C3⋊(C3⋊D3)12yescomplemented
115C2×C3C2×D3×C3⋊D3113yes—
116C2×C3C22×C3⋊(C3⋊D3)11yescharacteristic
117D3—952no—
118C22—272no—
119C22—274no—
120C22D3×C3⋊(C3⋊D3)11yescharacteristic
121C4—32no—
122C4—812no—
123C3—213no—
124C3C2×Dic3×C3⋊D3113yescomplemented
125C3C2×C4×C3⋊(C3⋊D3)11yescharacteristic, complemented
126C2—274no—
127C2Dic3×C3⋊(C3⋊D3)12yesdirect product factor
128C2C2×D3×C3⋊(C3⋊D3)11yescharacteristic
subgroupG/H
center, hypercenterC22D3×C3⋊(C3⋊D3)
commutator, hypocenterC34C22×C4
FrattiniC2C2×D3×C3⋊(C3⋊D3)
Fitting, socleC22×C34C22
prime radicalC22×C3×C3⋊(C3⋊D3)C2
cosocleC2×C34C23
LCSG ▷ C34
UCSC1 ◁ C22

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×C3⋊(C3⋊(C3⋊Dic3))C2semidirect
3. C2×C33×Dic3C2semidirect
4. Dic3×C3⋊(C3⋊D3)C2direct (N × H)
8. C2×C3×C3⋊(C3⋊D3)C4semidirect
10. C3⋊(C3⋊(C3⋊Dic3))C22semidirect
11. C33×Dic3C22semidirect
19. C2×C32×Dic3D3semidirect
21. C2×C34C2×C4semidirect
23. C3×C3⋊(C3⋊D3)C2×C4semidirect
29. C2×C3⋊(C3⋊D3)Dic3direct (N × H)
37. C32×Dic3C2×D3semidirect
38. C34C22×C4semidirect
46. C2×C3×Dic3C3⋊D3semidirect
50. C2×C33C4×D3semidirect
51. C2×C33C2×Dic3semidirect
54. C3⋊(C3⋊D3)C2×Dic3direct (N × H)
69. C3×Dic3C2×C3⋊D3semidirect
71. C33C2×C4×D3semidirect
72. C33C22×Dic3semidirect
80. C2×Dic3C3⋊(C3⋊D3)direct (N × H)
84. C2×C32D3×Dic3semidirect
85. C2×C32C4×C3⋊D3semidirect
102. Dic3C2×C3⋊(C3⋊D3)direct (N × H)
104. C32C2×D3×Dic3semidirect
105. C32C2×C4×C3⋊D3semidirect
113. C2×C3Dic3×C3⋊D3semidirect
114. C2×C3C4×C3⋊(C3⋊D3)semidirect
124. C3C2×Dic3×C3⋊D3semidirect
125. C3C2×C4×C3⋊(C3⋊D3)semidirect
127. C2Dic3×C3⋊(C3⋊D3)direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C22×C3×C3⋊(C3⋊D3)2
C2×C3⋊(C3⋊(C3⋊Dic3))2
C2×C33×Dic32
Dic3×C3⋊(C3⋊D3)2
C2×C4×C3⋊(C3⋊D3)3
C2×Dic3×C3⋊D33