Finite Group Databasebeta

← C2×#72234951 #72261740

Subgroups — C2×#72234951 #72261740

Order 3072.

At least 9212 conjugacy classes of subgroups; only the 227 normal subgroups are shown.

normal subgroups only

This group has at least 9212 conjugacy classes of subgroups — too many to draw. Only its 227 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×#10586813C311yesmaximal, characteristic, complemented, Sylow
2C2×#11710658C211yesmaximal, characteristic, complemented
3#72234951C212yesmaximal, direct product factor
4#11710658C2211yescharacteristic, complemented
5#10586813C2×C312yescomplemented
6C2×#62694C2×C311yescharacteristic, complemented
7C2×#3549A413yescomplemented
8C2×#3567A416yescomplemented
9C2×#3532A416yescomplemented
10#62694C22×C311yescharacteristic, complemented
11C22×H(3,𝔽4)A411yescharacteristic, complemented
12C2×#3444A413yescomplemented
13C23×#127A412yescomplemented
14#3567C2×A4112yescomplemented
15#3549C2×A416yescomplemented
16#3532C2×A4112yescomplemented
17C2×#564C2×A416yescomplemented
18C2×H(3,𝔽4)C2×A411yescharacteristic
19C2×H(3,𝔽4)C2×A412yescomplemented
20C2×H(3,𝔽4)C2×A413yescomplemented
21C2×H(3,𝔽4)C2×A416yescomplemented
22#3444C2×A416yescomplemented
23C22×#127C2×A414yescomplemented
24C23×C42C2×A412yescomplemented
25C23×C42C2×A413yescomplemented
26C26C22⋊A412yes—
27#564C22×A416yescomplemented
28H(3,𝔽4)C22×A411yescharacteristic
29H(3,𝔽4)C22×A416yescomplemented
30H(3,𝔽4)C22×A413yescomplemented
31C2×#127C22⋊A411yescharacteristic
32C2×#127C22⋊A416yes—
33C2×#126C22⋊A416yes—
34C2×#120C22⋊A413yes—
35C22×C42C22×A412yescomplemented
36C22×C42C22×A413yescomplemented
37C22×C42C22⋊A413yes—
38C25C2×C22⋊A412yes—
39C25C2×C22⋊A413yes—
40C25C2×C22⋊A414yes—
41#127C2×C22⋊A412yes—
42#127C2×C22⋊A4112yes—
43#126C2×C22⋊A4112yes—
44#120C2×C22⋊A416yes—
45C2×C42C2×C22⋊A416yes—
46C2×C42C2×C22⋊A416yes—
47C2×C42C2×C22⋊A416yes—
48C2×C42C2×C22⋊A413yes—
49C2×C42C2×C22⋊A411yescharacteristic
50C24C22×C22⋊A412yes—
51C24C22×C22⋊A413yes—
52C24C22⋊(C22⋊A4)11yescharacteristic
53C42C22×C22⋊A411yescharacteristic
54C42C22×C22⋊A413yes—
55C42C22×C22⋊A416yes—
56C42C22×C22⋊A416yes—
57C23C2×C22⋊(C22⋊A4)11yescharacteristic
58C23C2×C22⋊(C22⋊A4)12yes—
59C22C22×C22⋊(C22⋊A4)11yescharacteristic
60C2#7223495111yescharacteristic, direct product factor
subgroupG/H
center, hypercenterC2#72234951
commutator, cosocle, hypocenter#62694C22×C3
FrattiniC22C22×C22⋊(C22⋊A4)
FittingC2×#10586813C3
socleC23C2×C22⋊(C22⋊A4)
LCSG ▷ #62694
UCSC1 ◁ C2

Complements

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

NH ≅ G/Ntype
1. C2×#10586813C3semidirect
2. C2×#11710658C2semidirect
3. #72234951C2direct (N × H)
4. #11710658C22semidirect
5. #10586813C2×C3semidirect
6. C2×#62694C2×C3semidirect
7. C2×#3549A4semidirect
8. C2×#3567A4semidirect
9. C2×#3532A4semidirect
10. #62694C22×C3semidirect
11. C22×H(3,𝔽4)A4semidirect
12. C2×#3444A4semidirect
13. C23×#127A4semidirect
14. #3567C2×A4semidirect
15. #3549C2×A4semidirect
16. #3532C2×A4semidirect
17. C2×#564C2×A4semidirect
19. C2×H(3,𝔽4)C2×A4semidirect
20. C2×H(3,𝔽4)C2×A4semidirect
21. C2×H(3,𝔽4)C2×A4semidirect
22. #3444C2×A4semidirect
23. C22×#127C2×A4semidirect
24. C23×C42C2×A4semidirect
25. C23×C42C2×A4semidirect
27. #564C22×A4semidirect
29. H(3,𝔽4)C22×A4semidirect
30. H(3,𝔽4)C22×A4semidirect
35. C22×C42C22×A4semidirect
36. C22×C42C22×A4semidirect
60. C2#72234951direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#105868133
C2×#117106582
#722349512
C2×#3549⋊C34
C2×#3567⋊C34
C2×#3532⋊C34
C22×#54644
C2×#3444⋊C34
C2×#856474