Finite Group Databasebeta

← #11724679

Subgroups — #11724679

Order 864.

2324 conjugacy classes of subgroups; only the 527 normal subgroups are shown.

normal subgroups only

This group has 2324 conjugacy classes of subgroups — too many to draw. Only its 527 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
1C2×C33×D4C211yesmaximal, characteristic, complemented
2C2×#6312C214yesmaximal, complemented
3#89078C214yesmaximal, complemented
4D4×C3⋊(C3⋊D3)C214yesmaximal, complemented
5#89076C212yesmaximal, complemented
6#64361—313nomaximal
7C23×C33C2212yescomplemented
8C22×C3⋊(C3⋊D3)C2214yescomplemented
9C33×D4C2214yescomplemented
10C2×C33×C4C2211yescharacteristic, complemented
11#6312C2214yescomplemented
12#6312C2218yescomplemented
13C2×C3⋊(C3⋊Dic3)C2214yescomplemented
14C4⋊(C3⋊(C3⋊D3))C2212yescomplemented
15C4×C3⋊(C3⋊D3)C2214yescomplemented
16#6308C2212yescomplemented
17C2×C32×D4D3113yescomplemented
18C2×#638—352no—
19#3830—352no—
20D4×C3⋊D3—352no—
21#3828—326no—
22C22×C33—42no—
23C22×C33C2314yes—
24C22×C33C2311yescharacteristic
25C2×C3⋊(C3⋊D3)—44no—
26C2×C3⋊(C3⋊D3)C2314yes—
27C33×C4C2312yes—
28C3⋊(C3⋊Dic3)C2314yes—
29#1004—913no—
30C23×C32C2×D3126yescomplemented
31C22×C3⋊D3—352no—
32C32×D4C2×D3152yescomplemented
33C2×C32×C4C2×D3113yescomplemented
34#638—352no—
35#638—3104no—
36C2×C3⋊Dic3—352no—
37C4⋊(C3⋊D3)—326no—
38C4×C3⋊D3—352no—
39#634—326no—
40C2×C33—24no—
41C2×C33—21no—
42C2×C33C2411yescharacteristic
43C3⋊(C3⋊D3)—24no—
44C2×C3×D4C3⋊D3113yescomplemented
45C2×#67—952no—
46#237—952no—
47D3×D4—952no—
48#235—926no—
49C22×C32—426no—
50C22×C32C22×D3152yes—
51C22×C32C22×D3113yes—
52C2×C3⋊D3—352no—
53C2×C3⋊D3—1252no—
54C32×C4C22×D3126yes—
55C3⋊Dic3—352no—
56H(5,𝔽2)—271noSylow
57C33H(5,𝔽2)11yescharacteristic, complemented, Sylow
58C23×C3C2×C3⋊D3126yescomplemented
59C22×D3—952no—
60C3×D4C2×C3⋊D3152yescomplemented
61C2×C3×C4C2×C3⋊D3113yescomplemented
62#67—952no—
63#67—9104no—
64C2×Dic3—952no—
65D12—926no—
66C4×D3—952no—
67Dic6—926no—
68C2×C32—252no—
69C2×C32—213no—
70C2×C32C23×D3113yes—
71C3⋊D3—652no—
72#41—274no—
73#41—272no—
74C2×D4—274no—
75C2×D4—274no—
76C2×D4C3⋊(C3⋊D3)11yescharacteristic, complemented
77C22×C3—426no—
78C22×C3C22×C3⋊D3152yes—
79C22×C3C22×C3⋊D3113yes—
80C2×D3—952no—
81C2×D3—3652no—
82C3×C4C22×C3⋊D3126yes—
83Dic3—952no—
84C32#1004113yescomplemented
85C23—274no—
86C23C2×C3⋊(C3⋊D3)12yescomplemented
87Q8—272no—
88D4—278no—
89D4—274no—
90D4—272no—
91D4C2×C3⋊(C3⋊D3)14yescomplemented
92C2×C4—274no—
93C2×C4—274no—
94C2×C4C2×C3⋊(C3⋊D3)11yescharacteristic, complemented
95C2×C3—252no—
96C2×C3—213no—
97C2×C3C23×C3⋊D3113yes—
98D3—1852no—
99C22—42no—
100C22—274no—
101C22—1084no—
102C22C22×C3⋊(C3⋊D3)14yes—
103C22C22×C3⋊(C3⋊D3)11yescharacteristic
104C4—274no—
105C4C22×C3⋊(C3⋊D3)12yes—
106C3#64361113yescomplemented
107C2—21no—
108C2—24no—
109C2—544no—
110C2C23×C3⋊(C3⋊D3)11yescharacteristic
subgroupG/H
center, FrattiniC2C23×C3⋊(C3⋊D3)
commutator, socle, cosocleC2×C33C24
FittingC2×C33×D4C2
hypercenterC2×D4C3⋊(C3⋊D3)
hypocenterC33H(5,𝔽2)
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C2×C33×D4C2semidirect
2. C2×#6312C2semidirect
3. #89078C2semidirect
4. D4×C3⋊(C3⋊D3)C2semidirect
5. #89076C2semidirect
7. C23×C33C22semidirect
8. C22×C3⋊(C3⋊D3)C22semidirect
9. C33×D4C22semidirect
10. C2×C33×C4C22semidirect
11. #6312C22semidirect
12. #6312C22semidirect
13. C2×C3⋊(C3⋊Dic3)C22semidirect
14. C4⋊(C3⋊(C3⋊D3))C22semidirect
15. C4×C3⋊(C3⋊D3)C22semidirect
16. #6308C22semidirect
17. C2×C32×D4D3semidirect
30. C23×C32C2×D3semidirect
32. C32×D4C2×D3semidirect
33. C2×C32×C4C2×D3semidirect
44. C2×C3×D4C3⋊D3semidirect
57. C33H(5,𝔽2)semidirect
58. C23×C3C2×C3⋊D3semidirect
60. C3×D4C2×C3⋊D3semidirect
61. C2×C3×C4C2×C3⋊D3semidirect
76. C2×D4C3⋊(C3⋊D3)semidirect
84. C32#1004semidirect
86. C23C2×C3⋊(C3⋊D3)semidirect
91. D4C2×C3⋊(C3⋊D3)semidirect
94. C2×C4C2×C3⋊(C3⋊D3)semidirect
106. C3#64361semidirect

Maximal subgroups

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

subgroupindex
C2×C33×D42
C2×#63122
#890782
D4×C3⋊(C3⋊D3)2
#890762
#643613