Finite Group Databasebeta

← #72272620

Subgroups — #72272620

Order 3072.

At least 13127 conjugacy classes of subgroups; only the 537 normal subgroups are shown.

normal subgroups only

This group has at least 13127 conjugacy classes of subgroups — too many to draw. Only its 537 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#72272614C216yesmaximal, complemented
2#72272580C311yesmaximal, characteristic, complemented, Sylow
3#72262063C216yesmaximal, complemented
4#72262057C213yesmaximal, complemented
5#11710845C2211yescharacteristic, complemented
6#63026⋊C3C2216yescomplemented
7#63011⋊C3C2213yescomplemented
8#11710781C2212yescomplemented
9#62973⋊C3C22112yescomplemented
10#62972⋊C3C2216yescomplemented
11#62841⋊C3C2213yescomplemented
12#11710731C2212yescomplemented
13#10586976C2×C313yescomplemented
14#10586965C2×C316yescomplemented
15#10586817C2×C316yescomplemented
16#3568⋊C3C2316yescomplemented
17#3549⋊C3C2313yescomplemented
18#3532⋊C3C2316yescomplemented
19#63040C22×C311yescharacteristic, complemented
20#63026C22×C316yescomplemented
21#63011C22×C313yescomplemented
22#62980C22×C312yescomplemented
23#62973C22×C3112yescomplemented
24#62972C22×C316yescomplemented
25#62841C22×C313yescomplemented
26#62817C22×C312yescomplemented
27C22×H(3,𝔽4)A412yescomplemented
28C22×#560A413yescomplemented
29#5464C2411yescharacteristic, complemented
30#3568C23×C316yescomplemented
31#3549C23×C313yescomplemented
32#3532C23×C316yescomplemented
33C2×H(3,𝔽4)C2×A416yes—
34C2×H(3,𝔽4)C2×A4112yescomplemented
35C2×H(3,𝔽4)C2×A416yescomplemented
36C2×#560C2×A4112yescomplemented
37C2×#560C2×A416yescomplemented
38C2×#560C2×A416yescomplemented
39C2×#560C2×A4112yescomplemented
40C22×#127C2×A413yescomplemented
41C22×#126C2×A416yescomplemented
42C22×#120C2×A416yescomplemented
43C26C22×A412yescomplemented
44C26C22⋊A411yescharacteristic
45H(3,𝔽4)C22×A412yes—
46H(3,𝔽4)C22×A416yes—
47H(3,𝔽4)C22×A4112yes—
48H(3,𝔽4)C22×A416yescomplemented
49H(3,𝔽4)C22×A412yescomplemented
50H(3,𝔽4)C22×A414yescomplemented
51H(3,𝔽4)C24×C311yescharacteristic, complemented
52#560C22×A416yes—
53#560C22×A4112yes—
54#560C22×A416yescomplemented
55#560C22×A416yescomplemented
56#560C22×A416yescomplemented
57#560C22×A4112yescomplemented
58C2×#127C22×A413yes—
59C2×#127C22×A4112yescomplemented
60C2×#127C22×A413yescomplemented
61C2×#126C22×A416yes—
62C2×#126C22×A4112yescomplemented
63C2×#126C22×A416yescomplemented
64C2×#126C22×A4112yescomplemented
65C2×#120C22×A4112yes—
66C2×#120C22×A416yes—
67C2×#120C22×A4112yescomplemented
68C2×#120C22×A416yescomplemented
69C22×C42C22×A413yescomplemented
70C25C23×A416yes—
71C25C2×C22⋊A416yes—
72C25C2×C22⋊A413yes—
73C25C2×C22⋊A416yes—
74#127C23×A416yes—
75#127C23×A416yes—
76#126C23×A4112yes—
77#126C23×A4112yes—
78#120C23×A416yes—
79#120C23×A416yes—
80#120C23×A4112yes—
81C2×C42C23×A416yes—
82C2×C42C23×A413yes—
83C24C24×A412yes—
84C24C22×C22⋊A416yes—
85C24C22×C22⋊A412yes—
86C24C22×C22⋊A416yes—
87C24C22×C22⋊A4112yes—
88C24C22×C22⋊A411yescharacteristic
89C24C22×C22⋊A413yes—
90C24C22×C22⋊A412yes—
91C24C22×C22⋊A413yes—
92C42C24×A413yes—
93C23C23×C22⋊A416yes—
94C23C23×C22⋊A416yes—
95C23C23×C22⋊A413yes—
96C22C24×C22⋊A411yescharacteristic
subgroupG/H
center, hypercenterC1G
commutator, cosocle, hypocenterH(3,𝔽4)C24×C3
Frattini, socleC22C24×C22⋊A4
Fitting#72272580C3
LCSG ▷ H(3,𝔽4)
UCSC1

Complements

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

NH ≅ G/Ntype
1. #72272614C2semidirect
2. #72272580C3semidirect
3. #72262063C2semidirect
4. #72262057C2semidirect
5. #11710845C22semidirect
6. #63026⋊C3C22semidirect
7. #63011⋊C3C22semidirect
8. #11710781C22semidirect
9. #62973⋊C3C22semidirect
10. #62972⋊C3C22semidirect
11. #62841⋊C3C22semidirect
12. #11710731C22semidirect
13. #10586976C2×C3semidirect
14. #10586965C2×C3semidirect
15. #10586817C2×C3semidirect
16. #3568⋊C3C23semidirect
17. #3549⋊C3C23semidirect
18. #3532⋊C3C23semidirect
19. #63040C22×C3semidirect
20. #63026C22×C3semidirect
21. #63011C22×C3semidirect
22. #62980C22×C3semidirect
23. #62973C22×C3semidirect
24. #62972C22×C3semidirect
25. #62841C22×C3semidirect
26. #62817C22×C3semidirect
27. C22×H(3,𝔽4)A4semidirect
28. C22×#560A4semidirect
29. #5464C24semidirect
30. #3568C23×C3semidirect
31. #3549C23×C3semidirect
32. #3532C23×C3semidirect
34. C2×H(3,𝔽4)C2×A4semidirect
35. C2×H(3,𝔽4)C2×A4semidirect
36. C2×#560C2×A4semidirect
37. C2×#560C2×A4semidirect
38. C2×#560C2×A4semidirect
39. C2×#560C2×A4semidirect
40. C22×#127C2×A4semidirect
41. C22×#126C2×A4semidirect
42. C22×#120C2×A4semidirect
43. C26C22×A4semidirect
48. H(3,𝔽4)C22×A4semidirect
49. H(3,𝔽4)C22×A4semidirect
50. H(3,𝔽4)C22×A4semidirect
51. H(3,𝔽4)C24×C3semidirect
54. #560C22×A4semidirect
55. #560C22×A4semidirect
56. #560C22×A4semidirect
57. #560C22×A4semidirect
59. C2×#127C22×A4semidirect
60. C2×#127C22×A4semidirect
62. C2×#126C22×A4semidirect
63. C2×#126C22×A4semidirect
64. C2×#126C22×A4semidirect
67. C2×#120C22×A4semidirect
68. C2×#120C22×A4semidirect
69. C22×C42C22×A4semidirect

Maximal subgroups

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

subgroupindex
#722726142
#722725803
#722620632
#722620572
C22×#54504
C22×#560⋊C34