Finite Group Databasebeta

← #72272647

Subgroups — #72272647

Order 3072.

At least 25314 conjugacy classes of subgroups; only the 226 normal subgroups are shown.

normal subgroups only

This group has at least 25314 conjugacy classes of subgroups — too many to draw. Only its 226 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#72272580C311yesmaximal, characteristic, complemented, Sylow
2#72235027C213yesmaximal, complemented
3C22⋊#5464C2211yescharacteristic, complemented
4#10586796C2×C313yescomplemented
5#63040A411yescharacteristic, complemented
6#63026A413yescomplemented
7#62980A412yescomplemented
8#62973A416yescomplemented
9#62839A413yescomplemented
10#62817A411yescharacteristic, complemented
11#62817A411yescharacteristic, complemented
12C2×#3532A413yescomplemented
13C22×H(3,𝔽4)A411yescharacteristic, complemented
14C22×H(3,𝔽4)C22×C311yescharacteristic, complemented
15#3568C2×A416yescomplemented
16#3568C2×A416yescomplemented
17#3568C2×A416yescomplemented
18#3549C2×A413yescomplemented
19#3549C2×A416yescomplemented
20#3532C2×A413yescomplemented
21#3532C2×A416yescomplemented
22#3532C2×A413yescomplemented
23#3532C2×A416yescomplemented
24C2×H(3,𝔽4)C2×A413yescomplemented
25#3444C2×A413yescomplemented
26#3444C2×A416yescomplemented
27C22×#120C2×A413yescomplemented
28C22×#120C2×A413yescomplemented
29C26C22×A411yescharacteristic
30C26C22×A411yescharacteristic, complemented
31C26C22⋊A411yescharacteristic, complemented
32H(3,𝔽4)C22×A413yes—
33H(3,𝔽4)C22×A411yescharacteristic
34H(3,𝔽4)C22×A416yescomplemented
35H(3,𝔽4)C22×A413yescomplemented
36H(3,𝔽4)C22×A411yescharacteristic, complemented
37H(3,𝔽4)C22×A412yescomplemented
38H(3,𝔽4)C22⋊A411yescharacteristic
39H(3,𝔽4)C22⋊A411yescharacteristic, complemented
40H(3,𝔽4)C22⋊A412yescomplemented
41H(3,𝔽4)C22⋊A411yescharacteristic, complemented
42#560C22⋊A413yescomplemented
43C2×#127C22⋊A413yescomplemented
44C2×#126C22⋊A416yescomplemented
45C2×#120C22⋊A413yescomplemented
46C22×C42C22×A413yescomplemented
47C25C2×C22⋊A413yes—
48C25C2×C22⋊A413yes—
49#127C2×C22⋊A416yes—
50#127C2×C22⋊A413yes—
51#126C2×C22⋊A416yes—
52#126C2×C22⋊A416yes—
53#126C2×C22⋊A416yes—
54#120C2×C22⋊A413yes—
55#120C2×C22⋊A416yes—
56#120C2×C22⋊A413yes—
57#120C2×C22⋊A413yes—
58#120C2×C22⋊A416yes—
59C2×C42C2×C22⋊A413yes—
60C2×C42C2×C22⋊A416yes—
61C24C22×C22⋊A412yes—
62C24C22×C22⋊A411yescharacteristic
63C24C22×C22⋊A411yescharacteristic
64C24C22×C22⋊A413yes—
65C24C22×C22⋊A411yescharacteristic
66C24C22×C22⋊A411yescharacteristic
67C24C22⋊(C22⋊A4)11yescharacteristic
68C42C22×C22⋊A413yes—
69C42C22×C22⋊A416yes—
70C42C22×C22⋊A413yes—
71C23C2×C22⋊(C22⋊A4)13yes—
72C22C22×C22⋊(C22⋊A4)11yescharacteristic
subgroupG/H
center, hypercenterC1G
commutator, cosocle, hypocenterC22×H(3,𝔽4)C22×C3
Frattini, socleC22C22×C22⋊(C22⋊A4)
Fitting#72272580C3
LCSG ▷ C22×H(3,𝔽4)
UCSC1

Complements

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

NH ≅ G/Ntype
1. #72272580C3semidirect
2. #72235027C2semidirect
3. C22⋊#5464C22semidirect
4. #10586796C2×C3semidirect
5. #63040A4semidirect
6. #63026A4semidirect
7. #62980A4semidirect
8. #62973A4semidirect
9. #62839A4semidirect
10. #62817A4semidirect
11. #62817A4semidirect
12. C2×#3532A4semidirect
13. C22×H(3,𝔽4)A4semidirect
14. C22×H(3,𝔽4)C22×C3semidirect
15. #3568C2×A4semidirect
16. #3568C2×A4semidirect
17. #3568C2×A4semidirect
18. #3549C2×A4semidirect
19. #3549C2×A4semidirect
20. #3532C2×A4semidirect
21. #3532C2×A4semidirect
22. #3532C2×A4semidirect
23. #3532C2×A4semidirect
24. C2×H(3,𝔽4)C2×A4semidirect
25. #3444C2×A4semidirect
26. #3444C2×A4semidirect
27. C22×#120C2×A4semidirect
28. C22×#120C2×A4semidirect
30. C26C22×A4semidirect
31. C26C22⋊A4semidirect
34. H(3,𝔽4)C22×A4semidirect
35. H(3,𝔽4)C22×A4semidirect
36. H(3,𝔽4)C22×A4semidirect
37. H(3,𝔽4)C22×A4semidirect
39. H(3,𝔽4)C22⋊A4semidirect
40. H(3,𝔽4)C22⋊A4semidirect
41. H(3,𝔽4)C22⋊A4semidirect
42. #560C22⋊A4semidirect
43. C2×#127C22⋊A4semidirect
44. C2×#126C22⋊A4semidirect
45. C2×#120C22⋊A4semidirect
46. C22×C42C22×A4semidirect

Maximal subgroups

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

subgroupindex
#722725803
#722350272
#117108454
#63026⋊C34
#117107814
#62973⋊C34
#62839⋊C34
#117107314
#117107294
C2×#3532⋊C34
#117106504