Finite Group Databasebeta

← #72241425

Subgroups — #72241425

Order 3072.

Only the 710 normal subgroups are shown; non-normal subgroups are omitted.

normal subgroups only

Only the group's 710 normal subgroups are shown; non-normal subgroups are omitted.

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

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#72241414C216yesmaximal, complemented
2#72241377C219yesmaximal, complemented
3#72227619C211yesmaximal, characteristic, complemented
4#72215767C216yesmaximal, complemented
5C2×#11715828C219yesmaximal, complemented
6#11715831C2212yescomplemented
7#11715830C2216yescomplemented
8#11715829C22118yescomplemented
9#11715828C22136yescomplemented
10#11715828C22118yescomplemented
11#11715827C22118yescomplemented
12C2×#87522C2219yescomplemented
13C2×#87516C22118yescomplemented
14C2×#87520C2219yescomplemented
15C22×#5935C2216yescomplemented
16#11710937C2216yescomplemented
17C2×#87547C2219yescomplemented
18H(9,𝔽2)D311yescharacteristic, complemented
19#87548C2312yescomplemented
20#87547C23118yescomplemented
21C2×#5950C2319yescomplemented
22C22×#997C2316yescomplemented
23#87522C23118yescomplemented
24#87520C2316yescomplemented
25#87520C23118yescomplemented
26#87518C2316yescomplemented
27#87516C23136yescomplemented
28C2×#5935C2319yescomplemented
29C2×#5935C23118yescomplemented
30C2×#5933C2319yescomplemented
31#63103C2×D316yescomplemented
32C2×H(7,𝔽2)C2×D319yescomplemented
33#5950C2416yes—
34C2×#997C2419yes—
35#5935C2411yescharacteristic, complemented
36#5935C2419yescomplemented
37#5933C2416yescomplemented
382−1+6S412yescomplemented
392−1+6C22×D312yescomplemented
40H(7,𝔽2)C22×D3118yescomplemented
41C2×#585C22×D319yescomplemented
42C22×H(5,𝔽2)S411yescharacteristic
43C22×H(5,𝔽2)C22×D316yescomplemented
44#997C2511yescharacteristic
45#585C2×S4112yescomplemented
46#585C23×D316yes—
47C2×2−1+4C2×S4118yescomplemented
48C2×H(5,𝔽2)C23×D319yes—
49C22×#41C2×S416yes—
50C23×D4C2×S419yes—
51C25C22×S416yes—
522−1+4C22×S4136yescomplemented
53H(5,𝔽2)C22×S414yescomplemented
54H(5,𝔽2)#102011yescharacteristic
55H(5,𝔽2)C24×D311yescharacteristic
56C2×#41C22×S4118yescomplemented
57C22×Q8C22×S412yes—
58C22×Q8C22×S4112yescomplemented
59C22×D4C22×S4118yes—
60C23×C4C22×S419yes—
61C24C23×S419yes—
62#41C23×S4112yes—
63#41C2×#102016yes—
64C2×Q8C23×S4118yes—
65C2×D4C2×#102019yes—
66C22×C4C23×S416yes—
67C23C24×S411yescharacteristic
68C23C22×#102016yes—
69Q8C24×S412yes—
70Q8C22×#102012yes—
71D4C22×#1020118yes—
72C2×C4C22×#102019yes—
73C22C23×#102019yes—
74C4C23×#102016yes—
75C2C24×#102011yescharacteristic
subgroupG/H
center, Frattini, socleC2C24×#1020
commutator, cosocle, hypocenter#997C25
FittingH(9,𝔽2)D3
hypercenterH(5,𝔽2)#1020
LCSG ▷ #997
UCSC1 ◁ C2 ◁ H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. #72241414C2semidirect
2. #72241377C2semidirect
3. #72227619C2semidirect
4. #72215767C2semidirect
5. C2×#11715828C2semidirect
6. #11715831C22semidirect
7. #11715830C22semidirect
8. #11715829C22semidirect
9. #11715828C22semidirect
10. #11715828C22semidirect
11. #11715827C22semidirect
12. C2×#87522C22semidirect
13. C2×#87516C22semidirect
14. C2×#87520C22semidirect
15. C22×#5935C22semidirect
16. #11710937C22semidirect
17. C2×#87547C22semidirect
18. H(9,𝔽2)D3semidirect
19. #87548C23semidirect
20. #87547C23semidirect
21. C2×#5950C23semidirect
22. C22×#997C23semidirect
23. #87522C23semidirect
24. #87520C23semidirect
25. #87520C23semidirect
26. #87518C23semidirect
27. #87516C23semidirect
28. C2×#5935C23semidirect
29. C2×#5935C23semidirect
30. C2×#5933C23semidirect
31. #63103C2×D3semidirect
32. C2×H(7,𝔽2)C2×D3semidirect
35. #5935C24semidirect
36. #5935C24semidirect
37. #5933C24semidirect
38. 2−1+6S4semidirect
39. 2−1+6C22×D3semidirect
40. H(7,𝔽2)C22×D3semidirect
41. C2×#585C22×D3semidirect
43. C22×H(5,𝔽2)C22×D3semidirect
45. #585C2×S4semidirect
47. C2×2−1+4C2×S4semidirect
52. 2−1+4C22×S4semidirect
53. H(5,𝔽2)C22×S4semidirect
56. C2×#41C22×S4semidirect
58. C22×Q8C22×S4semidirect

Maximal subgroups

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

subgroupindex
#722414142
#722413772
#722413013
#722276192
#722157672
C2×#117158282
#117157974
S4×H(5,𝔽2)4