Finite Group Databasebeta

← C2×#72341141 #72341254

Subgroups — C2×#72341141 #72341254

Order 6144.

At least 8692 conjugacy classes of subgroups; only the 178 normal subgroups are shown.

normal subgroups only

This group has at least 8692 conjugacy classes of subgroups — too many to draw. Only its 178 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×#72341137C211yesmaximal, characteristic, complemented
2C2×#72261513C212yesmaximal, complemented
3C2×#72310694C211yesmaximal, characteristic, complemented
4C2×#11714271C2211yescharacteristic, complemented
5#72341141C218yesmaximal, direct product factor
6#72341137C2214yescomplemented
7C2×#72340506C211yesmaximal, characteristic, complemented
8C2×#34524⋊C3C2211yescharacteristic, complemented
9#72340506C2214yescomplemented
10C2×#8075294D311yescharacteristic, complemented
11C22×#11715753C212yesmaximal, complemented
12C22×#3532⋊C3C2212yescomplemented
13C22×#85559C2211yescharacteristic
14C22×#85559C2212yescomplemented
15#72310694C2214yescomplemented
16#72261513C2218yescomplemented
17C2×#11715753C2218yescomplemented
18C2×#85559D414yescomplemented
19C2×#85559C2312yes—
20C2×#85559C2314yescomplemented
21#11714271C2312yescomplemented
22C2×#3532⋊C3C2314yescomplemented
23C22×#5464C2311yescharacteristic
24#34524⋊C3C2312yescomplemented
25C22×#3532C2×D312yescomplemented
26#8075294C2×D314yescomplemented
27C2×#34524C2×D311yescharacteristic, complemented
28C2×#5464C2×D412yescomplemented
29C2×#5464C2411yescharacteristic
30#85559C2×D414yescomplemented
31C2×#3532C22×D314yescomplemented
32C22×H(3,𝔽4)C22×D311yescharacteristic
33C2×#2867S411yescharacteristic, complemented
34C2×#2846S412yescomplemented
35#34524C22×D312yescomplemented
36#5464C22×D411yescharacteristic, complemented
37C27C2×S412yescomplemented
38C2×H(3,𝔽4)D3×D412yescomplemented
39C2×H(3,𝔽4)C23×D311yescharacteristic
40C22×#127C2×S412yescomplemented
41C22×#120C2×S412yes—
42#2867C2×S414yescomplemented
43#2846C2×S418yescomplemented
44C2×#379C2×S412yescomplemented
45C2×C43C2×S411yescharacteristic, complemented
46C26C22×S412yes—
47C26C22×S414yescomplemented
48C23×D4#102011yescharacteristic
49H(3,𝔽4)C2×D3×D411yescharacteristic, complemented
50C2×#127C22×S414yescomplemented
51C2×#120C22×S414yes—
52C22×C42C22×S411yescharacteristic
53#379C22×S414yescomplemented
54C43C22×S412yescomplemented
55C25D4×S414yes—
56C25C23×S412yes—
57C25C2×#102012yes—
58C22×D4C2×#102014yes—
59C23×C4C2×#102011yescharacteristic
60C2×C42D4×S412yes—
61C2×C42C23×S411yescharacteristic
62C24C2×D4×S412yes—
63C24C22×#102014yes—
64C24C22×#102011yescharacteristic
65C22×C4C22×#102012yes—
66C42C2×D4×S411yescharacteristic
67C23D4×#102012yes—
68C23C23×#102011yescharacteristic
69C22C2×D4×#102011yescharacteristic
70C22#7221885711yescharacteristic
71C2C2×#7221885711yescharacteristic
72C2#7234114112yesdirect product factor
subgroupG/H
center, hypercenterC22#72218857
commutator, cosocleC2×#5464C24
FrattiniC23C23×#1020
FittingC2×#8075294D3
socleC24C22×#1020
hypocenter#5464C22×D4
LCSG ▷ C2×#5464 ▷ #5464
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
1. C2×#72341137C2semidirect
2. C2×#72261513C2semidirect
3. C2×#72310694C2semidirect
4. C2×#11714271C22semidirect
5. #72341141C2direct (N × H)
6. #72341137C22semidirect
7. C2×#72340506C2semidirect
8. C2×#34524⋊C3C22semidirect
9. #72340506C22semidirect
10. C2×#8075294D3semidirect
11. C22×#11715753C2semidirect
12. C22×#3532⋊C3C22semidirect
14. C22×#85559C22semidirect
15. #72310694C22semidirect
16. #72261513C22semidirect
17. C2×#11715753C22semidirect
18. C2×#85559D4semidirect
20. C2×#85559C23semidirect
21. #11714271C23semidirect
22. C2×#3532⋊C3C23semidirect
24. #34524⋊C3C23semidirect
25. C22×#3532C2×D3semidirect
26. #8075294C2×D3semidirect
27. C2×#34524C2×D3semidirect
28. C2×#5464C2×D4semidirect
30. #85559C2×D4semidirect
31. C2×#3532C22×D3semidirect
33. C2×#2867S4semidirect
34. C2×#2846S4semidirect
35. #34524C22×D3semidirect
36. #5464C22×D4semidirect
37. C27C2×S4semidirect
38. C2×H(3,𝔽4)D3×D4semidirect
40. C22×#127C2×S4semidirect
42. #2867C2×S4semidirect
43. #2846C2×S4semidirect
44. C2×#379C2×S4semidirect
45. C2×C43C2×S4semidirect
47. C26C22×S4semidirect
49. H(3,𝔽4)C2×D3×D4semidirect
50. C2×#127C22×S4semidirect
53. #379C22×S4semidirect
54. C43C22×S4semidirect
72. C2#72341141direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#723411372
C2×#722615132
C2×#723106942
#723411412
C2×#723405062
C2×#723127923
C22×#117157532
C2×#117136284
C2×#117132004