Finite Group Databasebeta

← #3449

Subgroups — #3449

Order 128.

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

normal subgroups only

This group has 778 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
1C22×#41C212yesmaximal, complemented
2#537C214yesmaximal, complemented
3#536C214yesmaximal, complemented
4#535C214yesmaximal, complemented
5#534C214yesmaximal, complemented
6#532C214yesmaximal, complemented
7#525C218yesmaximal, complemented
8C2×#117C211yesmaximal, characteristic, complemented
9C2×#41—216no—
10C2×#41C2214yes—
11C2×#41C2214yes—
12C2×#41C2214yescomplemented
13C2×#41C2214yescomplemented
14C22×Q8C2212yescomplemented
15C22×D4C2214yes—
16C22×D4C2212yescomplemented
17C23×C4C2211yescharacteristic
18C23×C4C2214yescomplemented
19C4⋊Q8C2214yescomplemented
20#127C2214yescomplemented
21#124C2218yescomplemented
22#123C22116yescomplemented
23C22⋊Q8C22116yescomplemented
24#121C22116yescomplemented
25#121C22116yescomplemented
26#120C22116yescomplemented
27C4×D4C22116yescomplemented
28#117C2214yescomplemented
29#117C2214yescomplemented
30C2×C4⋊C4C2212yescomplemented
31C2×C22⋊C4C2212yes—
32C2×C42C2212yescomplemented
33C24C2314yes—
34C24C2311yescharacteristic
35#41—216no—
36#41—216no—
37#41—216no—
38#41—216no—
39C2×Q8—28no—
40C2×Q8C2314yes—
41C2×Q8C2314yes—
42C2×D4—216no—
43C2×D4—28no—
44C2×D4C2318yes—
45C2×D4C2318yes—
46C2×D4C2314yes—
47C2×D4C2314yes—
48C2×D4C23116yes—
49C22×C4—216no—
50C22×C4D418yescomplemented
51C22×C4C2318yes—
52C22×C4C2314yes—
53C22×C4C2318yes—
54C22×C4C2314yes—
55C22×C4C2311yescharacteristic
56C22×C4C2312yes—
57C22×C4C2311yescharacteristic
58C22×C4C2312yes—
59C4⋊C4C2318yes—
60C4⋊C4C23116yes—
61C22⋊C4C2318yes—
62C22⋊C4C23132yes—
63C42C2314yes—
64C42C2314yes—
65C23—28no—
66C23—42no—
67C23—48no—
68C23C2×D414yes—
69C23C2418yes—
70C23C2412yes—
71C23C2411yescharacteristic
72Q8—28no—
73Q8—28no—
74D4—216no—
75D4—216no—
76D4—28no—
77D4—28no—
78D4—416no—
79C2×C4—28no—
80C2×C4—216no—
81C2×C4—216no—
82C2×C4—44no—
83C2×C4C2×D418yes—
84C2×C4C2×D418yes—
85C2×C4C2×D414yes—
86C2×C4C2×D414yes—
87C2×C4C2418yes—
88C2×C4C2418yes—
89C2×C4C2411yescharacteristic
90C2×C4C2412yes—
91C2×C4C2411yescharacteristic
92C22—22no—
93C22—24no—
94C22—28no—
95C22—216no—
96C22—44no—
97C22—416no—
98C22C22×D414yes—
99C22C22×D412yes—
100C22C2511yescharacteristic
101C4—28no—
102C4—28no—
103C4C22×D412yes—
104C4C22×D414yes—
105C4C22×D412yes—
106C2—24no—
107C2—22no—
108C2—48no—
109C2C23×D411yescharacteristic
110C2#58512yes—
subgroupG/H
centerC2×C4C24
commutator, Frattini, socle, cosocleC22C25
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ G
LECSG ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#41C2semidirect
2. #537C2semidirect
3. #536C2semidirect
4. #535C2semidirect
5. #534C2semidirect
6. #532C2semidirect
7. #525C2semidirect
8. C2×#117C2semidirect
12. C2×#41C22semidirect
13. C2×#41C22semidirect
14. C22×Q8C22semidirect
16. C22×D4C22semidirect
18. C23×C4C22semidirect
19. C4⋊Q8C22semidirect
20. #127C22semidirect
21. #124C22semidirect
22. #123C22semidirect
23. C22⋊Q8C22semidirect
24. #121C22semidirect
25. #121C22semidirect
26. #120C22semidirect
27. C4×D4C22semidirect
28. #117C22semidirect
29. #117C22semidirect
30. C2×C4⋊C4C22semidirect
32. C2×C42C22semidirect
50. C22×C4D4semidirect

Maximal subgroups

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

subgroupindex
C22×#412
#5372
#5362
#5352
#5342
#5322
#5252
C2×#1172