Finite Group Databasebeta

← #3485

Subgroups — #3485

Order 128.

780 conjugacy classes of subgroups; only the 428 normal subgroups are shown.

normal subgroups only

This group has 780 conjugacy classes of subgroups — too many to draw. Only its 428 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
1C2×H(5,𝔽2)C211yesmaximal, characteristic, complemented
2C22×#41C211yesmaximal, characteristic, complemented
3#547C218yesmaximal, complemented
4#546C218yesmaximal, complemented
5#536C212yesmaximal, complemented
6#534C214yesmaximal, complemented
7#525C212yesmaximal, complemented
8C2×#123C214yesmaximal, complemented
9#518C211yesmaximal, characteristic, complemented
10H(5,𝔽2)—28no—
11C2×#41—28no—
12C2×#41C2212yes—
13C2×#41C2218yescomplemented
14C2×#41C2214yescomplemented
15C22×Q8C2211yescharacteristic, complemented
16C22×D4C2214yes—
17C22×D4C2211yescharacteristic
18C22×D4C2212yescomplemented
19C22×D4C2214yescomplemented
20C23×C4C2212yescomplemented
21C23×C4C2211yescharacteristic, complemented
22C4⋊Q8C2214yescomplemented
23#124C2214yescomplemented
24#123C2214yescomplemented
25#123C2218yescomplemented
26#123C22116yescomplemented
27#123C2218yescomplemented
28C22⋊Q8C2214yescomplemented
29C22⋊Q8C2218yescomplemented
30#121C2214yescomplemented
31#121C22116yescomplemented
32#120C2214yescomplemented
33#120C2218yescomplemented
34C4×D4C2218yescomplemented
35C4×D4C2218yescomplemented
36#117C2212yescomplemented
37C2×C4⋊C4C2218yescomplemented
38C2×C22⋊C4C2214yescomplemented
39C2×C22⋊C4C2218yescomplemented
40C24C2314yes—
41C24C2311yescharacteristic
42C24C2312yes—
43#41—232no—
44#41—216no—
45#41—28no—
46C2×Q8—24no—
47C2×Q8C2314yes—
48C2×Q8C2312yes—
49C2×D4—216no—
50C2×D4—28no—
51C2×D4—216no—
52C2×D4D418yescomplemented
53C2×D4C2318yes—
54C2×D4C2314yes—
55C2×D4C2318yes—
56C2×D4C2312yes—
57C2×D4C2318yes—
58C2×D4C2314yes—
59C22×C4—28no—
60C22×C4—24no—
61C22×C4C2318yes—
62C22×C4C2314yes—
63C22×C4C2314yes—
64C22×C4C2314yes—
65C22×C4C2311yescharacteristic
66C22×C4C2312yes—
67C22×C4C2318yes—
68C22×C4C2311yescharacteristic
69C4⋊C4C2318yes—
70C4⋊C4C23116yes—
71C4⋊C4C2314yes—
72C22⋊C4C2314yes—
73C22⋊C4C2318yes—
74C22⋊C4C2318yes—
75C22⋊C4C23116yes—
76C22⋊C4C2318yes—
77C42C2314yes—
78C23—22no—
79C23—28no—
80C23—44no—
81C23—42no—
82C23—48no—
83C23C2×D418yes—
84C23C2414yes—
85C23C2412yes—
86C23C2411yescharacteristic
87C23C2414yes—
88Q8—28no—
89Q8—24no—
90D4—28no—
91D4—216no—
92D4—216no—
93D4—24no—
94D4—48no—
95D4C2×D4116yes—
96C2×C4—22no—
97C2×C4—24no—
98C2×C4—216no—
99C2×C4—28no—
100C2×C4—28no—
101C2×C4—28no—
102C2×C4—48no—
103C2×C4C2×D414yes—
104C2×C4C2418yes—
105C2×C4C2412yes—
106C2×C4C2414yes—
107C2×C4C2414yes—
108C2×C4C2412yes—
109C22—28no—
110C22—22no—
111C22—22no—
112C22—24no—
113C22—24no—
114C22—28no—
115C22—48no—
116C22—44no—
117C22—88no—
118C22C22×D412yes—
119C22C22×D418yes—
120C22C2511yescharacteristic
121C4—22no—
122C4—24no—
123C4—24no—
124C4—28no—
125C4C22×D414yes—
126C2—22no—
127C2—28no—
128C2—42no—
129C2—44no—
130C2C23×D411yescharacteristic
131C2#58512yes—
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. C2×H(5,𝔽2)C2semidirect
2. C22×#41C2semidirect
3. #547C2semidirect
4. #546C2semidirect
5. #536C2semidirect
6. #534C2semidirect
7. #525C2semidirect
8. C2×#123C2semidirect
9. #518C2semidirect
13. C2×#41C22semidirect
14. C2×#41C22semidirect
15. C22×Q8C22semidirect
18. C22×D4C22semidirect
19. C22×D4C22semidirect
20. C23×C4C22semidirect
21. C23×C4C22semidirect
22. C4⋊Q8C22semidirect
23. #124C22semidirect
24. #123C22semidirect
25. #123C22semidirect
26. #123C22semidirect
27. #123C22semidirect
28. C22⋊Q8C22semidirect
29. C22⋊Q8C22semidirect
30. #121C22semidirect
31. #121C22semidirect
32. #120C22semidirect
33. #120C22semidirect
34. C4×D4C22semidirect
35. C4×D4C22semidirect
36. #117C22semidirect
37. C2×C4⋊C4C22semidirect
38. C2×C22⋊C4C22semidirect
39. C2×C22⋊C4C22semidirect
52. C2×D4D4semidirect

Maximal subgroups

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

subgroupindex
C2×H(5,𝔽2)2
C22×#412
#5472
#5462
#5362
#5342
#5252
C2×#1232
#5182