Finite Group Databasebeta

← #62817

Subgroups — #62817

Order 256.

4747 conjugacy classes of subgroups; only the 2859 normal subgroups are shown.

normal subgroups only

This group has 4747 conjugacy classes of subgroups — too many to draw. Only its 2859 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#3532C2124yesmaximal, complemented
2#3490C2118yesmaximal, complemented
3#3484C2118yesmaximal, complemented
4C22×#120C213yesmaximal, complemented
5C26C2211yescharacteristic, complemented
6C2×H(5,𝔽2)C2216yescomplemented
7C23×D4C2219yescomplemented
8H(3,𝔽4)C2218yescomplemented
9#560C22124yescomplemented
10#551C22172yescomplemented
11#546C221144yescomplemented
12D42C22172yescomplemented
13#543C2216yescomplemented
14#538C22172yescomplemented
15#535C22136yescomplemented
16#534C22118yescomplemented
17#534C22136yescomplemented
18#525C22136yescomplemented
19C2×#120C22136yescomplemented
20C2×#120C22136yescomplemented
21C2×#120C22112yescomplemented
22#518C22118yescomplemented
23C22×C22⋊C4C2219yescomplemented
24C25—412no—
25C25C2319yes—
26C25C2316yes—
27C25C2313yescomplemented
28H(5,𝔽2)—424no—
29C2×#41C23136yescomplemented
30C22×D4—436no—
31C22×D4C23118yes—
32C22×D4C23118yescomplemented
33C22×D4C23172yescomplemented
34C22×D4C23136yescomplemented
35C23×C4C2319yescomplemented
36#127C23124yescomplemented
37#126C23148yescomplemented
38#124C23172yescomplemented
39#123C23172yescomplemented
40#123C23172yescomplemented
41C22⋊Q8C23172yescomplemented
42#121C231144yescomplemented
43#121C23172yescomplemented
44#120C23172yes—
45#120C23124yescomplemented
46#120C23172yescomplemented
47#120C23124yescomplemented
48#120C23172yescomplemented
49#120C23124yescomplemented
50C4×D4C231144yescomplemented
51#117C23136yescomplemented
52C2×C22⋊C4C23118yes—
53C2×C22⋊C4C23118yescomplemented
54C2×C22⋊C4C23172yescomplemented
55C2×C22⋊C4C23136yescomplemented
56C24—236no—
57C24—436no—
58C24—436no—
59C24—418no—
60C24—464no—
61C24C2413yes—
62C24C24118yes—
63C24C2419yes—
64C24C24118yes—
65C24C2412yes—
66C24C2413yes—
67C24C2414yescomplemented
68#41—472no—
69C2×Q8C24112yescomplemented
70C2×D4—272no—
71C2×D4—436no—
72C2×D4—4144no—
73C2×D4—472no—
74C2×D4C24172yes—
75C2×D4C24172yes—
76C2×D4C24136yescomplemented
77C22×C4—418no—
78C22×C4C24118yes—
79C22×C4C24136yes—
80C22×C4C24136yescomplemented
81C4⋊C4C24172yescomplemented
82C22⋊C4C24172yes—
83C22⋊C4C24172yes—
84C22⋊C4C24172yescomplemented
85C42C24124yescomplemented
86C23—218no—
87C23—272no—
88C23—218no—
89C23—236no—
90C23—46no—
91C23—418no—
92C23—418no—
93C23—46no—
94C23—436no—
95C23—412no—
96C23—4144no—
97C23—496no—
98C23—824no—
99C23H(5,𝔽2)112yescomplemented
100C23C2516yes—
101C23C25112yes—
102C23C2519yes—
103Q8—412no—
104D4—2144no—
105D4—472no—
106D4—436no—
107C2×C4—236no—
108C2×C4—436no—
109C2×C4—436no—
110C2×C4C25136yes—
111C22—224no—
112C22—218no—
113C22—236no—
114C22—218no—
115C22—412no—
116C22—472no—
117C22—436no—
118C22—48no—
119C22—836no—
120C22—164no—
121C22C2×H(5,𝔽2)118yes—
122C22C2611yescharacteristic
123C4—236no—
124C2—218no—
125C2—46no—
126C2—412no—
127C2C22×H(5,𝔽2)13yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC22C26
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ G
LECSG ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #3532C2semidirect
2. #3490C2semidirect
3. #3484C2semidirect
4. C22×#120C2semidirect
5. C26C22semidirect
6. C2×H(5,𝔽2)C22semidirect
7. C23×D4C22semidirect
8. H(3,𝔽4)C22semidirect
9. #560C22semidirect
10. #551C22semidirect
11. #546C22semidirect
12. D42C22semidirect
13. #543C22semidirect
14. #538C22semidirect
15. #535C22semidirect
16. #534C22semidirect
17. #534C22semidirect
18. #525C22semidirect
19. C2×#120C22semidirect
20. C2×#120C22semidirect
21. C2×#120C22semidirect
22. #518C22semidirect
23. C22×C22⋊C4C22semidirect
27. C25C23semidirect
29. C2×#41C23semidirect
32. C22×D4C23semidirect
33. C22×D4C23semidirect
34. C22×D4C23semidirect
35. C23×C4C23semidirect
36. #127C23semidirect
37. #126C23semidirect
38. #124C23semidirect
39. #123C23semidirect
40. #123C23semidirect
41. C22⋊Q8C23semidirect
42. #121C23semidirect
43. #121C23semidirect
45. #120C23semidirect
46. #120C23semidirect
47. #120C23semidirect
48. #120C23semidirect
49. #120C23semidirect
50. C4×D4C23semidirect
51. #117C23semidirect
53. C2×C22⋊C4C23semidirect
54. C2×C22⋊C4C23semidirect
55. C2×C22⋊C4C23semidirect
67. C24C24semidirect
69. C2×Q8C24semidirect
76. C2×D4C24semidirect
80. C22×C4C24semidirect
81. C4⋊C4C24semidirect
84. C22⋊C4C24semidirect
85. C42C24semidirect
99. C23H(5,𝔽2)semidirect

Maximal subgroups

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

subgroupindex
#35322
#34902
#34842
C22×#1202