Finite Group Databasebeta

← #3496

Subgroups — #3496

Order 128.

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

normal subgroups only

This group has 830 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
1C2×H(5,𝔽2)C212yesmaximal, complemented
2#547C214yesmaximal, complemented
3#546C218yesmaximal, complemented
4D42C214yesmaximal, complemented
5#537C212yesmaximal, complemented
6#534C214yesmaximal, complemented
7#532C212yesmaximal, complemented
8C2×#121C214yesmaximal, complemented
9C2×C4×D4C211yesmaximal, characteristic, complemented
10H(5,𝔽2)—216no—
11C2×#41C2214yescomplemented
12C2×#41C2218yescomplemented
13C22×D4C2211yescharacteristic
14C22×D4C2218yescomplemented
15C22×D4C2218yescomplemented
16C22×D4C2214yescomplemented
17C23×C4C2212yescomplemented
18C4⋊Q8C2212yescomplemented
19#127C2212yescomplemented
20#124C2214yescomplemented
21#123C22116yescomplemented
22C22⋊Q8C2218yescomplemented
23#121C2214yescomplemented
24#121C2218yescomplemented
25#121C2218yescomplemented
26#121C2218yescomplemented
27#121C2214yescomplemented
28#121C2218yescomplemented
29#120C22116yescomplemented
30C4×D4C2214yescomplemented
31C4×D4C2214yescomplemented
32C4×D4C2214yescomplemented
33C4×D4C2214yescomplemented
34C2×C4⋊C4C2211yescharacteristic, complemented
35C2×C4⋊C4C2214yescomplemented
36C2×C22⋊C4C2212yescomplemented
37C2×C22⋊C4C2218yescomplemented
38C2×C42C2211yescharacteristic, complemented
39C24C2318yes—
40C24C2312yes—
41#41—232no—
42#41—216no—
43C2×Q8C2314yes—
44C2×D4—232no—
45C2×D4—232no—
46C2×D4—48no—
47C2×D4D418yescomplemented
48C2×D4C2314yes—
49C2×D4C23116yes—
50C2×D4C2318yes—
51C2×D4C2314yes—
52C2×D4C2314yes—
53C2×D4C2318yes—
54C2×D4C2314yes—
55C22×C4—44no—
56C22×C4C2311yescharacteristic
57C22×C4C2318yes—
58C22×C4C2314yes—
59C22×C4C2318yes—
60C22×C4C2311yescharacteristic
61C22×C4C2314yes—
62C22×C4C2312yes—
63C22×C4C2311yescharacteristic
64C4⋊C4C2314yes—
65C4⋊C4C2314yes—
66C4⋊C4C2318yes—
67C4⋊C4C2314yes—
68C22⋊C4C2318yes—
69C22⋊C4C2318yes—
70C22⋊C4C23116yes—
71C22⋊C4C2318yes—
72C42C2312yes—
73C42C2312yes—
74C23—216no—
75C23—42no—
76C23—42no—
77C23—48no—
78C23—48no—
79C23C2×D418yescomplemented
80C23C2418yes—
81C23C2414yes—
82C23C2411yescharacteristic
83Q8—28no—
84D4—232no—
85D4—216no—
86D4—28no—
87D4—28no—
88D4—48no—
89D4—44no—
90D4C2×D4116yescomplemented
91C2×C4—216no—
92C2×C4—216no—
93C2×C4—22no—
94C2×C4—28no—
95C2×C4—44no—
96C2×C4—42no—
97C2×C4—41no—
98C2×C4C2×D414yescomplemented
99C2×C4C2412yes—
100C2×C4C2411yescharacteristic
101C2×C4C2414yes—
102C2×C4C2418yes—
103C2×C4C2412yes—
104C2×C4C2411yescharacteristic
105C22—21no—
106C22—24no—
107C22—21no—
108C22—24no—
109C22—28no—
110C22—28no—
111C22—28no—
112C22—48no—
113C22—816no—
114C22C22×D418yes—
115C22C22×D412yes—
116C22C2511yescharacteristic
117C4—28no—
118C4—21no—
119C4—24no—
120C4—22no—
121C4C22×D414yes—
122C4H(5,𝔽2)12yescomplemented
123C2—22no—
124C2—28no—
125C2—48no—
126C2C23×D411yescharacteristic
127C2C2×H(5,𝔽2)11yescharacteristic
128C2#58511yescharacteristic
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. #547C2semidirect
3. #546C2semidirect
4. D42C2semidirect
5. #537C2semidirect
6. #534C2semidirect
7. #532C2semidirect
8. C2×#121C2semidirect
9. C2×C4×D4C2semidirect
11. C2×#41C22semidirect
12. C2×#41C22semidirect
14. C22×D4C22semidirect
15. C22×D4C22semidirect
16. C22×D4C22semidirect
17. C23×C4C22semidirect
18. C4⋊Q8C22semidirect
19. #127C22semidirect
20. #124C22semidirect
21. #123C22semidirect
22. C22⋊Q8C22semidirect
23. #121C22semidirect
24. #121C22semidirect
25. #121C22semidirect
26. #121C22semidirect
27. #121C22semidirect
28. #121C22semidirect
29. #120C22semidirect
30. C4×D4C22semidirect
31. C4×D4C22semidirect
32. C4×D4C22semidirect
33. C4×D4C22semidirect
34. C2×C4⋊C4C22semidirect
35. C2×C4⋊C4C22semidirect
36. C2×C22⋊C4C22semidirect
37. C2×C22⋊C4C22semidirect
38. C2×C42C22semidirect
47. C2×D4D4semidirect
79. C23C2×D4semidirect
90. D4C2×D4semidirect
98. C2×C4C2×D4semidirect
122. C4H(5,𝔽2)semidirect

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
#5472
#5462
D422
#5372
#5342
#5322
C2×#1212
C2×C4×D42