Finite Group Databasebeta

← C2×#72215886 #72229142

Subgroups — C2×#72215886 #72229142

Order 6144.

At least 11280 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has at least 11280 conjugacy classes of subgroups — too many to draw. Only its 126 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×#72215885C211yesmaximal, characteristic, complemented
2C2×#72215879C211yesmaximal, characteristic, complemented
3C2×#72215875C211yesmaximal, characteristic, complemented
4C2×#72209642C211yesmaximal, characteristic, complemented
5C2×#72215860C211yesmaximal, characteristic, complemented
6C2×#11714267C2211yescharacteristic, complemented
7C2×#72215851C211yesmaximal, characteristic, complemented
8C2×#8541⋊C3C2211yescharacteristic, complemented
9C2×#11709314C2211yescharacteristic, complemented
10C2×#72215837C211yesmaximal, characteristic, complemented
11C2×#11710729C2211yescharacteristic, complemented
12C2×#87588⋊C2C2211yescharacteristic
13C2×#87588⋊C2C2211yescharacteristic, complemented
14C2×#87588⋊C2C2211yescharacteristic, complemented
15#72215886C218yesmaximal, direct product factor
16#72215885C2214yescomplemented
17#72215879C2214yescomplemented
18#72215875C2214yescomplemented
19#72215860C2214yescomplemented
20#72215851C2214yescomplemented
21#72215837C2214yescomplemented
22C2×#347042D311yescharacteristic, complemented
23#72209642C2214yescomplemented
24C2×#87588D412yescomplemented
25#87588⋊C2D412yescomplemented
26#87588⋊C2C2312yes—
27#87588⋊C2C2312yescomplemented
28#87588⋊C2C2312yescomplemented
29#11714267C2312yescomplemented
30#11710729C2312yescomplemented
31C2×#85645C2311yescharacteristic
32#11709314C2312yescomplemented
33#8541⋊C3C2312yescomplemented
34C2×#62817C2×D311yescharacteristic, complemented
35C2×#15947C2×D311yescharacteristic, complemented
36#347042C2×D314yescomplemented
37C2×#8541C2×D311yescharacteristic, complemented
38C2×C22⋊(C22⋊A4)C2×D411yescharacteristic, complemented
39#87588C2×D414yescomplemented
40#85645C2×D411yescharacteristic, complemented
41#85645C2411yescharacteristic
42#62817C22×D312yescomplemented
43C23×#120C22×D311yescharacteristic
44C2×#2846S411yescharacteristic, complemented
45#15947C22×D312yescomplemented
46#8541C22×D312yescomplemented
47C22⋊(C22⋊A4)C22×D411yescharacteristic, complemented
48C27D3×D411yescharacteristic, complemented
49C27C2×S411yescharacteristic, complemented
50C22×#120D3×D411yescharacteristic, complemented
51C22×#120C2×S411yescharacteristic
52C22×#120C23×D311yescharacteristic
53#2846C2×S414yescomplemented
54C2×#379C2×S411yescharacteristic, complemented
55C26C2×D3×D411yescharacteristic, complemented
56C26C22×S411yescharacteristic
57C26C22×S412yescomplemented
58C2×#120C22×S412yes—
59#379C22×S412yescomplemented
60C25#1020⋊C211yescharacteristic, complemented
61C25D4×S411yescharacteristic
62C25D4×S411yescharacteristic
63C25C23×S411yescharacteristic
64C24C2×#1020⋊C211yescharacteristic
65C24C2×#1020⋊C212yescomplemented
66C24C2×D4×S411yescharacteristic
67C23#1171320011yescharacteristic
68C23#1171320011yescharacteristic
69C23C22×#1020⋊C211yescharacteristic
70C22C2×#1171320011yescharacteristic
71C2#7221588611yescharacteristic, direct product factor
subgroupG/H
center, hypercenterC2#72215886
commutator, cosocle#85645C24
FrattiniC25C23×S4
FittingC2×#347042D3
socleC23#11713200
hypocenterC22⋊(C22⋊A4)C22×D4
LCSG ▷ #85645 ▷ C22⋊(C22⋊A4)
UCSC1 ◁ C2

Complements

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

NH ≅ G/Ntype
1. C2×#72215885C2semidirect
2. C2×#72215879C2semidirect
3. C2×#72215875C2semidirect
4. C2×#72209642C2semidirect
5. C2×#72215860C2semidirect
6. C2×#11714267C22semidirect
7. C2×#72215851C2semidirect
8. C2×#8541⋊C3C22semidirect
9. C2×#11709314C22semidirect
10. C2×#72215837C2semidirect
11. C2×#11710729C22semidirect
13. C2×#87588⋊C2C22semidirect
14. C2×#87588⋊C2C22semidirect
15. #72215886C2direct (N × H)
16. #72215885C22semidirect
17. #72215879C22semidirect
18. #72215875C22semidirect
19. #72215860C22semidirect
20. #72215851C22semidirect
21. #72215837C22semidirect
22. C2×#347042D3semidirect
23. #72209642C22semidirect
24. C2×#87588D4semidirect
25. #87588⋊C2D4semidirect
27. #87588⋊C2C23semidirect
28. #87588⋊C2C23semidirect
29. #11714267C23semidirect
30. #11710729C23semidirect
32. #11709314C23semidirect
33. #8541⋊C3C23semidirect
34. C2×#62817C2×D3semidirect
35. C2×#15947C2×D3semidirect
36. #347042C2×D3semidirect
37. C2×#8541C2×D3semidirect
38. C2×C22⋊(C22⋊A4)C2×D4semidirect
39. #87588C2×D4semidirect
40. #85645C2×D4semidirect
42. #62817C22×D3semidirect
44. C2×#2846S4semidirect
45. #15947C22×D3semidirect
46. #8541C22×D3semidirect
47. C22⋊(C22⋊A4)C22×D4semidirect
48. C27D3×D4semidirect
49. C27C2×S4semidirect
50. C22×#120D3×D4semidirect
53. #2846C2×S4semidirect
54. C2×#379C2×S4semidirect
55. C26C2×D3×D4semidirect
57. C26C22×S4semidirect
59. #379C22×S4semidirect
60. C25#1020⋊C2semidirect
65. C24C2×#1020⋊C2semidirect
71. C2#72215886direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#722158852
C2×#722158792
C2×#722158752
C2×#722096422
C2×#722158602
C2×#722158512
C2×#722158372
C2×B(5,𝔽2)3
#722158862
C2×#117132004