Finite Group Databasebeta

← #3191

Subgroups — #3191

Order 128.

212 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 86 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#525C212yesmaximal, complemented
2#485C212yesmaximal, complemented
3#482C212yesmaximal, complemented
4#467C211yesmaximal, characteristic, complemented
5#466C211yesmaximal, characteristic, complemented
6#465C212yesmaximal, complemented
7#452C212yesmaximal, complemented
8#449C212yesmaximal, complemented
9C2×C22⋊C8C211yesmaximal, characteristic, complemented
10C2×#41C2212yes—
11C23×C4C2211yescharacteristic
12C2×Q16C2211yescharacteristic, complemented
13C2×SD16C2212yescomplemented
14C2×D8C2211yescharacteristic, complemented
15C22×C8C2211yescharacteristic
16C22×C8C2211yescharacteristic
17#123—22no—
18C22⋊Q8C2212yes—
19C22⋊Q8C2212yes—
20#121C2212yes—
21#121C2212yes—
22#120—22no—
23C4×D4—22no—
24C4×D4—22no—
25#117C2212yes—
26#107C2211yescharacteristic, complemented
27#107C2211yescharacteristic, complemented
28#106C2212yescomplemented
29#103C2212yescomplemented
30#103C2212yescomplemented
31#102C2212yescomplemented
32#102C2212yescomplemented
33C22⋊C8C2212yes—
34C22⋊C8C2212yes—
35C24D411yescharacteristic
36#41—42no—
37#41—42no—
38C2×Q8C2312yes—
39C2×D4—22no—
40C2×D4—22no—
41C2×D4—22no—
42C2×D4C2312yes—
43C22×C4—22no—
44C22×C4—41no—
45C22×C4—41no—
46C22×C4D411yescharacteristic
47C22×C4D412yes—
48C22×C4C2312yes—
49C22×C4C2311yescharacteristic
50Q16—41no—
51SD16—42no—
52D8—41no—
53C2×C8—41no—
54C2×C8—41no—
55C2×C8C2311yescharacteristic
56C2×C8C2312yes—
57C2×C8C2311yescharacteristic
58C4⋊C4—22no—
59C4⋊C4—22no—
60C4⋊C4C2312yes—
61C4⋊C4C2312yes—
62C22⋊C4—22no—
63C22⋊C4—22no—
64C22⋊C4—22no—
65C22⋊C4—22no—
66C22⋊C4—22no—
67C42—22no—
68C23—22no—
69C23—22no—
70C23—41no—
71C23—41no—
72C23C2×D412yes—
73C23C2×D411yescharacteristic
74Q8—22no—
75Q8—22no—
76D4—22no—
77D4—22no—
78D4—44no—
79D4—44no—
80D4—44no—
81C2×C4—21no—
82C2×C4—21no—
83C2×C4—22no—
84C2×C4—22no—
85C2×C4—22no—
86C2×C4—22no—
87C2×C4—22no—
88C2×C4—41no—
89C2×C4—42no—
90C2×C4—44no—
91C2×C4C2×D412yes—
92C2×C4C2×D411yescharacteristic
93C2×C4C2411yescharacteristic
94C8—21no—
95C8—22no—
96C8—21no—
97C22—21no—
98C22—21no—
99C22—22no—
100C22—22no—
101C22—22no—
102C22—44no—
103C22—42no—
104C22—42no—
105C22—42no—
106C22#13512yes—
107C22C22×D411yescharacteristic
108C4—22no—
109C4—21no—
110C4—42no—
111C4—42no—
112C4—42no—
113C4H(5,𝔽2)11yescharacteristic
114C4H(5,𝔽2)11yescharacteristic
115C2—22no—
116C2—42no—
117C2—82no—
118C2#53411yescharacteristic
119C2C2×#13511yescharacteristic
120C2#57511yescharacteristic
subgroupG/H
center, socleC22C22×D4
commutator, Frattini, cosocleC2×C4C24
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #525C2semidirect
2. #485C2semidirect
3. #482C2semidirect
4. #467C2semidirect
5. #466C2semidirect
6. #465C2semidirect
7. #452C2semidirect
8. #449C2semidirect
9. C2×C22⋊C8C2semidirect
12. C2×Q16C22semidirect
13. C2×SD16C22semidirect
14. C2×D8C22semidirect
26. #107C22semidirect
27. #107C22semidirect
28. #106C22semidirect
29. #103C22semidirect
30. #103C22semidirect
31. #102C22semidirect
32. #102C22semidirect

Maximal subgroups

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

subgroupindex
#5252
#4852
#4822
#4672
#4662
#4652
#4522
#4492
C2×C22⋊C82