Finite Group Databasebeta

← C2×#72231618 #72274616

Subgroups — C2×#72231618 #72274616

Order 9216.

At least 8302 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has at least 8302 conjugacy classes of subgroups — too many to draw. Only its 88 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×#72230753C211yesmaximal, characteristic, complemented
2C2×#10974249D311yescharacteristic, complemented
3C2×#72230842C211yesmaximal, characteristic, complemented
4C2×C3×#11715850C211yesmaximal, characteristic, complemented
5C2×C3×C22⋊#997C2211yescharacteristic, complemented
6#72231618C214yesmaximal, direct product factor
7#72230842C2212yescomplemented
8C3×#11715850C2212yescomplemented
9#72230753C2212yescomplemented
10C2×#11715850D311yescharacteristic, complemented
11C3×C22⋊#997C2311yescharacteristic, complemented
12#11715850C2×D312yescomplemented
13C2×C22⋊#997C2×D311yescharacteristic, complemented
14#10974249C2×D312yescomplemented
15C23×C3×H(5,𝔽2)C2×D311yescharacteristic, complemented
16C26×D3S411yescharacteristic, complemented
17C22×C3×H(5,𝔽2)C22×D311yescharacteristic, complemented
18C22⋊#997C22×D311yescharacteristic, complemented
19C22×#953S412yescomplemented
20C23×H(5,𝔽2)D3211yescharacteristic, complemented
21C26×C3C2×S411yescharacteristic
22C26×C3C2×S412yescomplemented
23C25×D3C2×S412yescomplemented
24C2×#953C2×S414yescomplemented
25C22×H(5,𝔽2)C2×D3211yescharacteristic, complemented
26C25×C3C22×S411yescharacteristic
27C25×C3C22×S412yescomplemented
28C24×D3#102011yescharacteristic
29C26D3×S411yescharacteristic
30C26D3×S412yescomplemented
31C24×C3#1020⋊C212yescomplemented
32C24×C3C2×#102011yescharacteristic
33C23×D3#593414yescomplemented
34C23×D3C2×#102012yes—
35C25C2×D3×S411yescharacteristic
36C25C2×D3×S412yescomplemented
37C23×C3C2×#1020⋊C212yescomplemented
38C23×C3C2×#593412yes—
39C23×C3C22×#102011yescharacteristic
40C22×D3C2×#593414yescomplemented
41C24#1059324412yescomplemented
42C24D3×#102011yescharacteristic
43C22×C3#87588⋊C211yescharacteristic
44C22×C3C22×#593411yescharacteristic
45C23C2×#1059324412yescomplemented
46C23D3×#593412yes—
47C23C2×D3×#102011yescharacteristic
48C2×C3#7220965312yescomplemented
49C2×C3C2×#87588⋊C211yescharacteristic
50C22C2×D3×#593411yescharacteristic
51C22#7223161111yescharacteristic
52C3C2×#7220965311yescharacteristic, complemented
53C2#7223161812yesdirect product factor
54C2C2×#7223161111yescharacteristic
subgroupG/H
center, hypercenterC22#72231611
commutator, cosocle, hypocenterC3×C22⋊#997C23
FrattiniC23C2×D3×#1020
FittingC23×C3×H(5,𝔽2)C2×D3
socleC24×C3C2×#1020
LCSG ▷ C3×C22⋊#997
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
1. C2×#72230753C2semidirect
2. C2×#10974249D3semidirect
3. C2×#72230842C2semidirect
4. C2×C3×#11715850C2semidirect
5. C2×C3×C22⋊#997C22semidirect
6. #72231618C2direct (N × H)
7. #72230842C22semidirect
8. C3×#11715850C22semidirect
9. #72230753C22semidirect
10. C2×#11715850D3semidirect
11. C3×C22⋊#997C23semidirect
12. #11715850C2×D3semidirect
13. C2×C22⋊#997C2×D3semidirect
14. #10974249C2×D3semidirect
15. C23×C3×H(5,𝔽2)C2×D3semidirect
16. C26×D3S4semidirect
17. C22×C3×H(5,𝔽2)C22×D3semidirect
18. C22⋊#997C22×D3semidirect
19. C22×#953S4semidirect
20. C23×H(5,𝔽2)D32semidirect
22. C26×C3C2×S4semidirect
23. C25×D3C2×S4semidirect
24. C2×#953C2×S4semidirect
25. C22×H(5,𝔽2)C2×D32semidirect
27. C25×C3C22×S4semidirect
30. C26D3×S4semidirect
31. C24×C3#1020⋊C2semidirect
33. C23×D3#5934semidirect
36. C25C2×D3×S4semidirect
37. C23×C3C2×#1020⋊C2semidirect
40. C22×D3C2×#5934semidirect
41. C24#10593244semidirect
45. C23C2×#10593244semidirect
48. C2×C3#72209653semidirect
52. C3C2×#72209653semidirect
53. C2#72231618direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#722307532
C2×#722315913
C22×#105932444
C2×#722308422
C2×C3×#117158502
#722316182
C2×#722096533
C22×D3×#10204