Finite Group Databasebeta

← H(3,𝔽4)×D181 #37547840

Subgroups — H(3,𝔽4)×D181 #37547840

Order 23168.

1408 conjugacy classes of subgroups; only the 459 normal subgroups are shown.

normal subgroups only

This group has 1408 conjugacy classes of subgroups — too many to draw. Only its 459 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
1#26042022C216yesmaximal, complemented
2#26042021C219yesmaximal, complemented
3#124×D181C219yesmaximal, complemented
4#120×D181C216yesmaximal, complemented
5H(3,𝔽4)×C181C211yesmaximal, characteristic, complemented
6C24×D181C2212yescomplemented
7C2×Q8×D181C2213yescomplemented
8C2×D4×D181C2219yes—
9#19153805C2219yescomplemented
10#19153804C2219yescomplemented
11#19153803C2219yescomplemented
12#19153802C22136yescomplemented
13#19153786C22118yescomplemented
14#19153785C2216yescomplemented
15#19153784C22118yescomplemented
16C22⋊C4×D181C22118yescomplemented
17C42×D181C2213yescomplemented
18#124×C181C2219yescomplemented
19#120×C181C2216yescomplemented
20C23×D181—412no—
21C23×D181C2316yes—
22C23×D181C2316yescomplemented
23C2×Dic362C2319yescomplemented
24Q8×D181—46no—
25C2×D724C2319yes—
26C2×C22⋊D181C23118yes—
27D4×D181—418no—
28C2×C4×D181C2319yes—
29C22⋊Dic181C23118yescomplemented
30#15668426C23136yescomplemented
31C4×Dic181C2319yescomplemented
32C24×C181C2312yescomplemented
33C2×Q8×C181C2313yescomplemented
34C2×D4×C181C2319yes—
35C22⋊C4×C181C23118yescomplemented
36C42×C181C2313yescomplemented
37C22×D181—236no—
38C22×D181—418no—
39C22×D181—432no—
40C22×D181C2416yes—
41C22×D181C2411yescharacteristic
42Dic362—49no—
43D724—49no—
44C22⋊D181—418no—
45C2×Dic181C2419yes—
46C4×D181—218no—
47C23×C181—46no—
48C23×C181C2416yes—
49Q8×C181—43no—
50D4×C181—49no—
51C2×C4×C181C2419yes—
52C22×C181—218no—
53C22×C181—48no—
54C22×C181C2511yescharacteristic
55C2×D181—218no—
56C2×D181—424no—
57C2×D181—424no—
58C2×D181H(5,𝔽2)16yes—
59C4×C181—29no—
60Dic181—29no—
61C2×C181—46no—
62C2×C181C2×H(5,𝔽2)13yes—
63D181—46no—
64D181H(3,𝔽4)14yesdirect product factor
65C181C2×H(3,𝔽4)11yescharacteristic, complemented, Sylow
66C2×H(3,𝔽4)—1811nomaximal, Sylow
67H(3,𝔽4)—1819no—
68H(3,𝔽4)—1816no—
69H(3,𝔽4)D18111yescharacteristic, direct product factor
70C2×#124—1819no—
71C2×#120—1816no—
72C25—1812no—
73C22×Q8—1813no—
74C22×D4—1819no—
75#124—1819no—
76#124—1819no—
77#124—18136no—
78#124—1819no—
79#124C2×D18119yescomplemented
80#120—1816no—
81#120—18118no—
82#120—18118no—
83#120C2×D18116yescomplemented
84C2×C22⋊C4—18118no—
85C2×C42—1813no—
86C24—1816no—
87C24—1816no—
88C24—72412no—
89C24C22×D18112yescomplemented
90C2×Q8—1819no—
91C2×Q8—7246no—
92C2×Q8C22×D18113yescomplemented
93C2×D4—18118no—
94C2×D4—1819no—
95C2×D4—72418no—
96C2×D4C22×D18119yes—
97C22×C4—1819no—
98C22⋊C4—18118no—
99C22⋊C4—18136no—
100C22⋊C4C22×D181118yescomplemented
101C42—1819no—
102C42C22×D18113yescomplemented
103C23—46no—
104C23—36236no—
105C23—1816no—
106C23—1811no—
107C23—72418no—
108C23—72432no—
109C23C23×D18116yes—
110Q8—43no—
111Q8—7249no—
112D4—49no—
113D4—7249no—
114D4—72418no—
115C2×C4—36218no—
116C2×C4—1819no—
117C2×C4C23×D18119yes—
118C22—218no—
119C22—48no—
120C22—36218no—
121C22—1816no—
122C22—72424no—
123C22—72424no—
124C22C24×D18111yescharacteristic
125C4—29no—
126C4—3629no—
127C2—46no—
128C2—1814no—
129C2—7246no—
130C2H(5,𝔽2)×D18113yes—
subgroupG/H
center, FrattiniC22C24×D181
commutator, socle, cosocleC22×C181C25
FittingH(3,𝔽4)×C181C2
hypercenterH(3,𝔽4)D181
hypocenterC181C2×H(3,𝔽4)
LCSG ▷ C22×C181 ▷ C181
UCSC1 ◁ C22 ◁ H(3,𝔽4)

Complements

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

NH ≅ G/Ntype
1. #26042022C2semidirect
2. #26042021C2semidirect
3. #124×D181C2semidirect
4. #120×D181C2semidirect
5. H(3,𝔽4)×C181C2semidirect
6. C24×D181C22semidirect
7. C2×Q8×D181C22semidirect
9. #19153805C22semidirect
10. #19153804C22semidirect
11. #19153803C22semidirect
12. #19153802C22semidirect
13. #19153786C22semidirect
14. #19153785C22semidirect
15. #19153784C22semidirect
16. C22⋊C4×D181C22semidirect
17. C42×D181C22semidirect
18. #124×C181C22semidirect
19. #120×C181C22semidirect
22. C23×D181C23semidirect
23. C2×Dic362C23semidirect
29. C22⋊Dic181C23semidirect
30. #15668426C23semidirect
31. C4×Dic181C23semidirect
32. C24×C181C23semidirect
33. C2×Q8×C181C23semidirect
35. C22⋊C4×C181C23semidirect
36. C42×C181C23semidirect
64. D181H(3,𝔽4)direct (N × H)
65. C181C2×H(3,𝔽4)semidirect
69. H(3,𝔽4)D181direct (N × H)
79. #124C2×D181semidirect
83. #120C2×D181semidirect
89. C24C22×D181semidirect
92. C2×Q8C22×D181semidirect
100. C22⋊C4C22×D181semidirect
102. C42C22×D181semidirect

Maximal subgroups

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

subgroupindex
#260420222
#260420212
#124×D1812
#120×D1812
H(3,𝔽4)×C1812
C2×H(3,𝔽4)181