Finite Group Databasebeta

← #3490

Subgroups — #3490

Order 128.

664 conjugacy classes of subgroups; only the 392 normal subgroups are shown.

normal subgroups only

This group has 664 conjugacy classes of subgroups — too many to draw. Only its 392 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#551C218yesmaximal, complemented
2#546C218yesmaximal, complemented
3#543C211yesmaximal, characteristic, complemented
4#538C214yesmaximal, complemented
5#534C211yesmaximal, characteristic, complemented
6#525C214yesmaximal, complemented
7C2×#120C212yesmaximal, complemented
8#518C212yesmaximal, complemented
9C22×C22⋊C4C211yesmaximal, characteristic, complemented
10C25C2211yescharacteristic
11C2×#41C2214yescomplemented
12C22×D4C2212yes—
13C22×D4C2214yescomplemented
14C23×C4C2212yescomplemented
15#126C2218yescomplemented
16#124C2218yescomplemented
17#123C2218yescomplemented
18#123C2218yescomplemented
19#123C2214yescomplemented
20C22⋊Q8C2214yescomplemented
21C22⋊Q8C2214yescomplemented
22C22⋊Q8C2218yescomplemented
23#121C2218yescomplemented
24#121C2214yescomplemented
25#121C2214yescomplemented
26#120C2218yes—
27#120C2214yescomplemented
28#120C2214yescomplemented
29#120C2214yescomplemented
30C4×D4C2218yescomplemented
31C4×D4C2218yescomplemented
32C4×D4C2218yescomplemented
33#117C2214yescomplemented
34#117C2214yescomplemented
35C2×C22⋊C4C2211yescharacteristic
36C2×C22⋊C4C2211yescharacteristic
37C2×C22⋊C4C2212yes—
38C2×C22⋊C4C2212yescomplemented
39C2×C22⋊C4C2218yescomplemented
40C2×C22⋊C4C2214yescomplemented
41C2×C22⋊C4C2214yescomplemented
42C24—24no—
43C24—44no—
44C24C2311yescharacteristic
45C24C2314yes—
46C24C2311yescharacteristic
47C24C2312yes—
48C24C2311yescharacteristic
49#41—48no—
50C2×Q8C2314yescomplemented
51C2×D4—28no—
52C2×D4—48no—
53C2×D4C2314yes—
54C2×D4C2314yes—
55C2×D4C2318yes—
56C2×D4C2318yes—
57C2×D4C2314yescomplemented
58C22×C4—44no—
59C22×C4C2312yes—
60C22×C4C2314yes—
61C22×C4C2312yes—
62C22×C4C2314yes—
63C22×C4C2312yes—
64C22×C4C2314yes—
65C22×C4C2318yescomplemented
66C4⋊C4C2318yescomplemented
67C4⋊C4C2314yescomplemented
68C4⋊C4C2314yescomplemented
69C4⋊C4C2318yescomplemented
70C22⋊C4C2314yes—
71C22⋊C4C2314yes—
72C22⋊C4C2314yes—
73C22⋊C4C2314yes—
74C22⋊C4C2318yes—
75C22⋊C4C2318yes—
76C22⋊C4C2318yes—
77C22⋊C4C2318yescomplemented
78C22⋊C4C2318yescomplemented
79C42C2318yescomplemented
80C23—22no—
81C23—24no—
82C23—28no—
83C23—22no—
84C23—28no—
85C23—24no—
86C23—42no—
87C23—44no—
88C23—42no—
89C23—44no—
90C23—416no—
91C23#4114yescomplemented
92C23C2412yes—
93C23C2414yes—
94C23C2412yes—
95C23C2411yescharacteristic
96C23C2412yes—
97Q8—44no—
98D4—28no—
99D4—28no—
100D4—216no—
101D4—48no—
102D4—44no—
103C2×C4—24no—
104C2×C4—28no—
105C2×C4—42no—
106C2×C4—44no—
107C2×C4—42no—
108C2×C4—44no—
109C2×C4—48no—
110C2×C4C2414yes—
111C2×C4C2414yes—
112C2×C4C2418yes—
113C2×C4C2414yes—
114C22—28no—
115C22—22no—
116C22—22no—
117C22—22no—
118C22—24no—
119C22—24no—
120C22—22no—
121C22—28no—
122C22—24no—
123C22—44no—
124C22—416no—
125C22—44no—
126C22—84no—
127C22C2×#4114yes—
128C22C2×#4112yes—
129C22H(5,𝔽2)14yes—
130C22C2511yescharacteristic
131C4—24no—
132C4—24no—
133C4—24no—
134C4—28no—
135C2—22no—
136C2—24no—
137C2—24no—
138C2—42no—
139C2—44no—
140C2C22×#4111yescharacteristic
141C2C2×H(5,𝔽2)12yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC22C25
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ G
LECSG ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #551C2semidirect
2. #546C2semidirect
3. #543C2semidirect
4. #538C2semidirect
5. #534C2semidirect
6. #525C2semidirect
7. C2×#120C2semidirect
8. #518C2semidirect
9. C22×C22⋊C4C2semidirect
11. C2×#41C22semidirect
13. C22×D4C22semidirect
14. C23×C4C22semidirect
15. #126C22semidirect
16. #124C22semidirect
17. #123C22semidirect
18. #123C22semidirect
19. #123C22semidirect
20. C22⋊Q8C22semidirect
21. C22⋊Q8C22semidirect
22. C22⋊Q8C22semidirect
23. #121C22semidirect
24. #121C22semidirect
25. #121C22semidirect
27. #120C22semidirect
28. #120C22semidirect
29. #120C22semidirect
30. C4×D4C22semidirect
31. C4×D4C22semidirect
32. C4×D4C22semidirect
33. #117C22semidirect
34. #117C22semidirect
38. C2×C22⋊C4C22semidirect
39. C2×C22⋊C4C22semidirect
40. C2×C22⋊C4C22semidirect
41. C2×C22⋊C4C22semidirect
50. C2×Q8C23semidirect
57. C2×D4C23semidirect
65. C22×C4C23semidirect
66. C4⋊C4C23semidirect
67. C4⋊C4C23semidirect
68. C4⋊C4C23semidirect
69. C4⋊C4C23semidirect
77. C22⋊C4C23semidirect
78. C22⋊C4C23semidirect
79. C42C23semidirect
91. C23#41semidirect

Maximal subgroups

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

subgroupindex
#5512
#5462
#5432
#5382
#5342
#5252
C2×#1202
#5182
C22×C22⋊C42