Finite Group Databasebeta

← #3280

Subgroups — #3280

Order 128.

263 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 263 conjugacy classes of subgroups — too many to draw. Only its 96 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×#135C211yesmaximal, characteristic, complemented
2C22×D8C211yesmaximal, characteristic, complemented
3#546C212yesmaximal, complemented
4#495C211yesmaximal, characteristic, complemented
5#467C211yesmaximal, characteristic, complemented
6#466C211yesmaximal, characteristic, complemented
7#463C212yesmaximal, complemented
8#450C212yesmaximal, complemented
9#449C212yesmaximal, complemented
10C4×D8C211yesmaximal, characteristic, complemented
11C8×D4C211yesmaximal, characteristic, complemented
12C2×#41C2212yescomplemented
13C22×D4C2212yes—
14#135—24no—
15C2×Q16C2211yescharacteristic, complemented
16C2×SD16C2212yescomplemented
17C2×SD16C2212yescomplemented
18C2×D8—24no—
19C2×D8C2212yes—
20C2×D8C2211yescharacteristic
21C2×D8C2211yescharacteristic
22C22×C8C2211yescharacteristic
23C22×C8C2211yescharacteristic, complemented
24#124C2212yescomplemented
25#123—22no—
26C22⋊Q8C2212yes—
27#121—22no—
28#121C2212yescomplemented
29#120—22no—
30C4×D4C2211yescharacteristic
31C4×D4C2212yescomplemented
32C2×C22⋊C4—22no—
33#107C2211yescharacteristic, complemented
34C4⋊C8C2211yescharacteristic, complemented
35#103C2212yescomplemented
36#102C2212yescomplemented
37#102C2212yescomplemented
38C22⋊C8C2211yescharacteristic
39C22⋊C8C2211yescharacteristic, complemented
40C4×C8C2211yescharacteristic, complemented
41C24—22no—
42#41—24no—
43#41—42no—
44C2×Q8C2312yes—
45C2×D4—24no—
46C2×D4—22no—
47C2×D4—22no—
48C2×D4—22no—
49C2×D4—42no—
50C2×D4D411yescharacteristic
51C2×D4C2312yes—
52C2×D4C2312yes—
53C22×C4—22no—
54C22×C4—22no—
55C22×C4C2311yescharacteristic
56C22×C4C2311yescharacteristic
57Q16—22no—
58SD16—24no—
59SD16—42no—
60D8—24no—
61D8—22no—
62D8D414yescomplemented
63C2×C8—22no—
64C2×C8—22no—
65C2×C8C2311yescharacteristic
66C2×C8C2311yescharacteristic
67C2×C8C2311yescharacteristic
68C2×C8C2311yescharacteristic
69C4⋊C4—22no—
70C4⋊C4D411yescharacteristic
71C4⋊C4C2312yes—
72C22⋊C4—22no—
73C22⋊C4—22no—
74C22⋊C4—22no—
75C22⋊C4—22no—
76C22⋊C4—22no—
77C22⋊C4D411yescharacteristic
78C22⋊C4D411yescharacteristic
79C42C2311yescharacteristic
80C23—22no—
81C23—22no—
82C23—44no—
83C23—44no—
84C23—82no—
85C23C2×D411yescharacteristic
86C23C2×D411yescharacteristic
87Q8—22no—
88Q8—22no—
89D4—22no—
90D4—22no—
91D4—22no—
92D4—22no—
93D4—44no—
94D4—44no—
95D4—82no—
96D4C2×D414yes—
97C2×C4—21no—
98C2×C4—21no—
99C2×C4—21no—
100C2×C4—21no—
101C2×C4—22no—
102C2×C4—22no—
103C2×C4—44no—
104C2×C4—82no—
105C2×C4C2×D411yescharacteristic
106C2×C4C2×D411yescharacteristic
107C2×C4C2×D411yescharacteristic
108C2×C4C2×D411yescharacteristic
109C2×C4C2411yescharacteristic
110C8—21no—
111C8—21no—
112C8—21no—
113C8C2×D412yes—
114C22—21no—
115C22—21no—
116C22—21no—
117C22—24no—
118C22—21no—
119C22—21no—
120C22—42no—
121C22—42no—
122C22—42no—
123C22—42no—
124C22—42no—
125C22—88no—
126C22#13512yescomplemented
127C22C22×D411yescharacteristic
128C4—21no—
129C4—21no—
130C4—21no—
131C4—21no—
132C4—42no—
133C4—42no—
134C4C22×D411yescharacteristic
135C4H(5,𝔽2)11yescharacteristic
136C2—22no—
137C2—41no—
138C2—44no—
139C2—82no—
140C2D4211yescharacteristic
141C2C2×#13511yescharacteristic
142C2#57611yescharacteristic
subgroupG/H
center, socleC22C22×D4
commutator, Frattini, cosocleC2×C4C24
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C4×D4 ◁ G
LECSG ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#135C2semidirect
2. C22×D8C2semidirect
3. #546C2semidirect
4. #495C2semidirect
5. #467C2semidirect
6. #466C2semidirect
7. #463C2semidirect
8. #450C2semidirect
9. #449C2semidirect
10. C4×D8C2semidirect
11. C8×D4C2semidirect
12. C2×#41C22semidirect
15. C2×Q16C22semidirect
16. C2×SD16C22semidirect
17. C2×SD16C22semidirect
23. C22×C8C22semidirect
24. #124C22semidirect
28. #121C22semidirect
31. C4×D4C22semidirect
33. #107C22semidirect
34. C4⋊C8C22semidirect
35. #103C22semidirect
36. #102C22semidirect
37. #102C22semidirect
39. C22⋊C8C22semidirect
40. C4×C8C22semidirect
62. D8D4semidirect
126. C22#135semidirect

Maximal subgroups

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

subgroupindex
C2×#1352
C22×D82
#5462
#4952
#4672
#4662
#4632
#4502
#4492
C4×D82
C8×D42