Finite Group Databasebeta

← #3501

Subgroups — #3501

Order 128.

550 conjugacy classes of subgroups; only the 430 normal subgroups are shown.

normal subgroups only

This group has 550 conjugacy classes of subgroups — too many to draw. Only its 430 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#554C214yesmaximal, complemented
2#554C218yesmaximal, complemented
3D4×Q8C214yesmaximal, complemented
4#544C212yesmaximal, complemented
5#543C214yesmaximal, complemented
6#533C212yesmaximal, complemented
7C2×C22⋊Q8C214yesmaximal, complemented
8#518C212yesmaximal
9C2×C4×D4C211yesmaximal, characteristic
10C22×Q8C2214yescomplemented
11C22×D4C2211yescharacteristic
12C23×C4C2212yes—
13C4⋊Q8C2212yescomplemented
14C4⋊Q8C2212yescomplemented
15C4⋊Q8C2218yescomplemented
16#125C2214yescomplemented
17#125C2218yescomplemented
18C22⋊Q8C2214yes—
19C22⋊Q8C2218yes—
20C22⋊Q8C2214yes—
21C22⋊Q8C2218yescomplemented
22C22⋊Q8C2218yescomplemented
23C22⋊Q8C22116yescomplemented
24C22⋊Q8C22116yescomplemented
25C4×Q8C2214yescomplemented
26C4×Q8C2214yescomplemented
27C4×D4C2214yes—
28C4×D4C2214yes—
29C4×D4C2218yes—
30C4×D4C2218yes—
31#117C2214yes—
32C2×C4⋊C4C2211yescharacteristic
33C2×C4⋊C4C2218yescomplemented
34C2×C4⋊C4C2214yescomplemented
35C2×C22⋊C4C2212yes—
36C2×C22⋊C4C2218yes—
37C2×C42C2211yescharacteristic
38C24C2312yes—
39C2×Q8—48no—
40C2×Q8C2314yes—
41C2×Q8C2314yes—
42C2×Q8C2318yes—
43C2×D4Q818yescomplemented
44C2×D4C2314yes—
45C22×C4—44no—
46C22×C4C2311yescharacteristic
47C22×C4C2314yes—
48C22×C4C2311yescharacteristic
49C22×C4C2312yes—
50C22×C4C2318yes—
51C22×C4C2318yes—
52C22×C4C2314yes—
53C22×C4C2311yescharacteristic
54C4⋊C4C2314yes—
55C4⋊C4C2318yes—
56C4⋊C4C2314yes—
57C4⋊C4C2318yes—
58C4⋊C4C23116yes—
59C4⋊C4C2314yes—
60C4⋊C4C2314yes—
61C4⋊C4C2318yes—
62C4⋊C4C2314yes—
63C22⋊C4C2318yes—
64C22⋊C4C2318yes—
65C22⋊C4C23116yes—
66C42C2312yes—
67C42C2312yes—
68C42C2318yes—
69C23—42no—
70C23—42no—
71C23C2×Q818yescomplemented
72C23C2411yescharacteristic
73C23C2414yes—
74Q8—28no—
75Q8—48no—
76Q8—44no—
77D4C2×Q8116yescomplemented
78C2×C4—22no—
79C2×C4—28no—
80C2×C4—44no—
81C2×C4—42no—
82C2×C4—48no—
83C2×C4—41no—
84C2×C4—48no—
85C2×C4C2×Q814yescomplemented
86C2×C4C2412yes—
87C2×C4C2411yescharacteristic
88C2×C4C2418yes—
89C2×C4C2418yes—
90C2×C4C2414yes—
91C2×C4C2412yes—
92C2×C4C2411yescharacteristic
93C22—24no—
94C22—21no—
95C22—24no—
96C22—21no—
97C22—48no—
98C22C22×Q812yes—
99C22C22×Q818yes—
100C22C2511yescharacteristic
101C4—21no—
102C4—24no—
103C4—28no—
104C4—28no—
105C4—22no—
106C4C22×Q814yes—
107C4H(5,𝔽2)12yes—
108C2—22no—
109C2—28no—
110C2C23×Q811yescharacteristic
111C2C2×H(5,𝔽2)11yescharacteristic
112C2#58511yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC22C25
prime radicalC22×D4C22
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ G
LECSG ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #554C2semidirect
2. #554C2semidirect
3. D4×Q8C2semidirect
4. #544C2semidirect
5. #543C2semidirect
6. #533C2semidirect
7. C2×C22⋊Q8C2semidirect
10. C22×Q8C22semidirect
13. C4⋊Q8C22semidirect
14. C4⋊Q8C22semidirect
15. C4⋊Q8C22semidirect
16. #125C22semidirect
17. #125C22semidirect
21. C22⋊Q8C22semidirect
22. C22⋊Q8C22semidirect
23. C22⋊Q8C22semidirect
24. C22⋊Q8C22semidirect
25. C4×Q8C22semidirect
26. C4×Q8C22semidirect
33. C2×C4⋊C4C22semidirect
34. C2×C4⋊C4C22semidirect
43. C2×D4Q8semidirect
71. C23C2×Q8semidirect
77. D4C2×Q8semidirect
85. C2×C4C2×Q8semidirect

Maximal subgroups

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

subgroupindex
#5542
D4×Q82
#5442
#5432
#5332
C2×C22⋊Q82
#5182
C2×C4×D42