Finite Group Databasebeta

← C2×#72341123 #72341250

Subgroups — C2×#72341123 #72341250

Order 6144.

At least 9595 conjugacy classes of subgroups; only the 218 normal subgroups are shown.

normal subgroups only

This group has at least 9595 conjugacy classes of subgroups — too many to draw. Only its 218 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×#72261513C214yesmaximal, complemented
2C2×#72310668C211yesmaximal, characteristic, complemented
3C2×#11714271C2212yescomplemented
4#72341123C218yesmaximal, direct product factor
5C22×#63011⋊C3C211yesmaximal, characteristic, complemented
6C22×#3532⋊C3C2212yescomplemented
7C22×#11715754C211yesmaximal, characteristic, complemented
8C22×#85559C2212yescomplemented
9C22×#3549⋊C3C2211yescharacteristic
10#72310668C2214yescomplemented
11#72261513C22116yescomplemented
12C2×#11715754C2214yescomplemented
13C22×#63011D311yescharacteristic, complemented
14C2×#63011⋊C3C2214yescomplemented
15C2×#85559C2314yescomplemented
16#11714271C2314yescomplemented
17C2×#3549⋊C3D414yescomplemented
18C2×#3549⋊C3C2312yes—
19C2×#3532⋊C3C2314yescomplemented
20C22×#5464C2311yescharacteristic
21C2×#63011C2×D314yescomplemented
22C22×#3549C2×D311yescharacteristic
23C22×#3532C2×D312yescomplemented
24#3549⋊C3C2×D414yescomplemented
25C2×#5464C2×D412yescomplemented
26C2×#5464C2411yescharacteristic
27C2×#3549#6714yescomplemented
28C2×#3549C22×D312yes—
29C2×#3532C22×D314yescomplemented
30C22×H(3,𝔽4)C22×D311yescharacteristic
31C23×#127S411yescharacteristic, complemented
32C23×#120S412yescomplemented
33#5464C22×D411yescharacteristic, complemented
34C27C2×S412yes—
35#3549C2×#6714yescomplemented
36C2×H(3,𝔽4)C2×#6712yescomplemented
37C2×H(3,𝔽4)C23×D311yescharacteristic
38C22×#127C2×S412yes—
39C22×#127C2×S414yescomplemented
40C22×#120C2×S412yes—
41C22×#120C2×S412yes—
42C22×#120C2×S418yescomplemented
43C23×C42C2×S411yescharacteristic
44C26C22×S414yes—
45C26C22×S412yes—
46C26#102011yescharacteristic
47H(3,𝔽4)C22×#6711yescharacteristic, complemented
48C2×#127C22×S414yes—
49C2×#120GL(2,ℤ/4)18yescomplemented
50C2×#120C22×S414yes—
51C2×#120C22×S414yes—
52C22×C42GL(2,ℤ/4)14yescomplemented
53C22×C42C22×S411yescharacteristic
54C22×C42C22×S412yes—
55C25C2×GL(2,ℤ/4)14yes—
56C25C23×S412yes—
57C25C2×#102012yes—
58C25C2×#102014yes—
59C25C2×#102011yescharacteristic
60#120C2×GL(2,ℤ/4)18yescomplemented
61C2×C42C2×GL(2,ℤ/4)12yes—
62C2×C42C2×GL(2,ℤ/4)14yescomplemented
63C2×C42C23×S411yescharacteristic
64C24C22×GL(2,ℤ/4)12yes—
65C24#8752414yes—
66C24C22×#102014yes—
67C24C22×#102011yescharacteristic
68C24C22×#102012yes—
69C42C22×GL(2,ℤ/4)11yescharacteristic
70C23C2×#8752412yes—
71C23C2×#8752414yes—
72C23C23×#102011yescharacteristic
73C22#7221885711yescharacteristic
74C22C22×#8752411yescharacteristic
75C2C2×#7221885711yescharacteristic
76C2#7234112312yesdirect product factor
subgroupG/H
center, hypercenterC22#72218857
commutator, cosocleC2×#5464C24
FrattiniC23C23×#1020
FittingC22×#63011D3
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×#72261513C2semidirect
2. C2×#72310668C2semidirect
3. C2×#11714271C22semidirect
4. #72341123C2direct (N × H)
5. C22×#63011⋊C3C2semidirect
6. C22×#3532⋊C3C22semidirect
7. C22×#11715754C2semidirect
8. C22×#85559C22semidirect
10. #72310668C22semidirect
11. #72261513C22semidirect
12. C2×#11715754C22semidirect
13. C22×#63011D3semidirect
14. C2×#63011⋊C3C22semidirect
15. C2×#85559C23semidirect
16. #11714271C23semidirect
17. C2×#3549⋊C3D4semidirect
19. C2×#3532⋊C3C23semidirect
21. C2×#63011C2×D3semidirect
23. C22×#3532C2×D3semidirect
24. #3549⋊C3C2×D4semidirect
25. C2×#5464C2×D4semidirect
27. C2×#3549#67semidirect
29. C2×#3532C22×D3semidirect
31. C23×#127S4semidirect
32. C23×#120S4semidirect
33. #5464C22×D4semidirect
35. #3549C2×#67semidirect
36. C2×H(3,𝔽4)C2×#67semidirect
39. C22×#127C2×S4semidirect
42. C22×#120C2×S4semidirect
47. H(3,𝔽4)C22×#67semidirect
49. C2×#120GL(2,ℤ/4)semidirect
52. C22×C42GL(2,ℤ/4)semidirect
60. #120C2×GL(2,ℤ/4)semidirect
62. C2×C42C2×GL(2,ℤ/4)semidirect
76. C2#72341123direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#722615132
C2×#723106682
#723411232
C2×#722519883
C22×#63011⋊C32
C22×#117157542
C22×#730844
C2×#117132014