Finite Group Databasebeta

← #72340822

Subgroups — #72340822

Order 6144.

At least 16380 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has at least 16380 conjugacy classes of subgroups — too many to draw. Only its 99 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#72340793C212yesmaximal, complemented
2#72262047C211yesmaximal, characteristic, complemented
3#72260975C2211yescharacteristic, complemented
4#72234660C214yesmaximal, complemented
5#72234218C2214yescomplemented
6C22×#63011D311yescharacteristic, complemented
7#72218927C2212yescomplemented
8C22⋊#5464C2311yescharacteristic, complemented
9C22×#3549C2×D311yescharacteristic, complemented
10C22×#3532C2×D312yescomplemented
11#63011S414yescomplemented
12C22×H(3,𝔽4)C22×D311yescharacteristic, complemented
13C23×#127S411yescharacteristic, complemented
14C23×#120S412yescomplemented
15C27C2×S412yescomplemented
16#3549C2×S414yescomplemented
17#3532C2×S418yescomplemented
18C22×#127C2×S412yescomplemented
19C22×#120C2×S412yes—
20C22×#120C2×S412yescomplemented
21C23×C42C2×S411yescharacteristic, complemented
22C26C22×S412yes—
23C26#102011yescharacteristic
24H(3,𝔽4)C22×S414yescomplemented
25C2×#127#102012yescomplemented
26C2×#120#102014yescomplemented
27C22×C42C22×S411yescharacteristic
28C25C2×#102011yescharacteristic
29C25C2×#102012yes—
30C25C2×#102014yescomplemented
31#127C2×#102014yescomplemented
32#120C2×#102014yes—
33#120C2×#102014yescomplemented
34C2×C42C2×#102012yescomplemented
35C24C22×#102011yescharacteristic
36C24C22×#102014yes—
37C24#8758811yescharacteristic
38C42C22×#102012yes—
39C23C2×#8758811yescharacteristic
40C23C2×#8758812yes—
41C22#7221885712yescomplemented
42C22C22×#8758811yescharacteristic
subgroupG/H
center, hypercenterC1G
commutator, cosocle, hypocenterC22⋊#5464C23
FrattiniC22C22×#87588
FittingC22×#63011D3
socleC24C22×#1020
LCSG ▷ C22⋊#5464
UCSC1

Complements

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

NH ≅ G/Ntype
1. #72340793C2semidirect
2. #72262047C2semidirect
3. #72260975C22semidirect
4. #72234660C2semidirect
5. #72234218C22semidirect
6. C22×#63011D3semidirect
7. #72218927C22semidirect
8. C22⋊#5464C23semidirect
9. C22×#3549C2×D3semidirect
10. C22×#3532C2×D3semidirect
11. #63011S4semidirect
12. C22×H(3,𝔽4)C22×D3semidirect
13. C23×#127S4semidirect
14. C23×#120S4semidirect
15. C27C2×S4semidirect
16. #3549C2×S4semidirect
17. #3532C2×S4semidirect
18. C22×#127C2×S4semidirect
20. C22×#120C2×S4semidirect
21. C23×C42C2×S4semidirect
24. H(3,𝔽4)C22×S4semidirect
25. C2×#127#1020semidirect
26. C2×#120#1020semidirect
30. C25C2×#1020semidirect
31. #127C2×#1020semidirect
33. #120C2×#1020semidirect
34. C2×C42C2×#1020semidirect
41. C22#72218857semidirect

Maximal subgroups

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

subgroupindex
#723407932
#722620472
#722520393
#722346602
C2×#117156904
#722188574
C2×#87588⋊C24