Finite Group Databasebeta

← #3295

Subgroups — #3295

Order 128.

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

normal subgroups only

This group has 231 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
1#554C211yesmaximal, characteristic, complemented
2D42C211yesmaximal, characteristic
3#486C211yesmaximal, characteristic, complemented
4#481C212yesmaximal, complemented
5#476C211yesmaximal, characteristic, complemented
6#465C212yesmaximal, complemented
7#460C211yesmaximal, characteristic, complemented
8#450C212yesmaximal, complemented
9C4×SD16C211yesmaximal, characteristic, complemented
10C8×D4C211yesmaximal, characteristic, complemented
11C2×#102C212yesmaximal, complemented
12C22×D4—21no—
13C22×D4C2212yes—
14C2×SD16C2211yescharacteristic
15C2×SD16C2212yescomplemented
16C22×C8C2212yescomplemented
17C4⋊Q8C2211yescharacteristic, complemented
18#127C2211yescharacteristic
19#125—21no—
20C22⋊Q8—22no—
21C22⋊Q8C2212yescomplemented
22#121—21no—
23#121C2212yes—
24#120—22no—
25C4×Q8C2211yescharacteristic, complemented
26C4×D4—21no—
27C4×D4C2211yescharacteristic
28C4×D4C2211yescharacteristic
29C2×C4⋊C4C2212yescomplemented
30#106C2211yescharacteristic, complemented
31#106C2212yescomplemented
32C4⋊C8C2211yescharacteristic, complemented
33#103C2211yescharacteristic, complemented
34#102C2211yescharacteristic
35#102C2212yes—
36#102C2212yescomplemented
37#102C2212yescomplemented
38#102C2212yescomplemented
39C22⋊C8C2212yescomplemented
40C4×C8C2211yescharacteristic, complemented
41C24—22no—
42C2×Q8—21no—
43C2×Q8C2311yescharacteristic
44C2×D4—24no—
45C2×D4—22no—
46C2×D4—22no—
47C2×D4—21no—
48C2×D4—21no—
49C2×D4—42no—
50C2×D4—44no—
51C2×D4D411yescharacteristic
52C2×D4C2312yes—
53C2×D4C2311yescharacteristic
54C22×C4—22no—
55C22×C4—21no—
56C22×C4C2312yes—
57SD16—22no—
58SD16—42no—
59C2×C8—42no—
60C2×C8C2311yescharacteristic
61C2×C8C2311yescharacteristic
62C2×C8C2312yes—
63C4⋊C4—22no—
64C4⋊C4—21no—
65C4⋊C4—21no—
66C4⋊C4—22no—
67C4⋊C4D411yescharacteristic
68C4⋊C4C2311yescharacteristic
69C4⋊C4C2312yes—
70C4⋊C4C2311yescharacteristic
71C22⋊C4—22no—
72C22⋊C4—21no—
73C22⋊C4—22no—
74C22⋊C4D412yes—
75C42—21no—
76C42C2311yescharacteristic
77C23—21no—
78C23—22no—
79C23—44no—
80C23—44no—
81C23—82no—
82C23C2×D412yes—
83Q8—21no—
84Q8—21no—
85Q8—81no—
86D4—22no—
87D4—21no—
88D4—22no—
89D4—44no—
90D4—44no—
91D4—42no—
92D4—81no—
93D4SD1614yescomplemented
94D4#4112yes—
95C2×C4—22no—
96C2×C4—22no—
97C2×C4—21no—
98C2×C4—21no—
99C2×C4—21no—
100C2×C4—22no—
101C2×C4—42no—
102C2×C4—82no—
103C2×C4C2×D411yescharacteristic
104C2×C4C2×D412yes—
105C2×C4C2×D411yescharacteristic
106C2×C4C2411yescharacteristic
107C8—21no—
108C8—21no—
109C8—22no—
110C22—22no—
111C22—22no—
112C22—22no—
113C22—41no—
114C22—41no—
115C22—42no—
116C22—42no—
117C22—42no—
118C22—88no—
119C22C2×SD1614yes—
120C22C22×D411yescharacteristic
121C4—22no—
122C4—21no—
123C4—41no—
124C4—41no—
125C4—41no—
126C4—42no—
127C4C2×SD1612yes—
128C4C2×#4111yescharacteristic
129C4H(5,𝔽2)11yescharacteristic
130C2—24no—
131C2—42no—
132C2—82no—
133C2#54611yescharacteristic
134C2C22×SD1611yescharacteristic
135C2#57611yescharacteristic
subgroupG/H
center, socleC22C22×D4
commutator, Frattini, cosocleC2×C4C24
prime radicalD42C2
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. #554C2semidirect
3. #486C2semidirect
4. #481C2semidirect
5. #476C2semidirect
6. #465C2semidirect
7. #460C2semidirect
8. #450C2semidirect
9. C4×SD16C2semidirect
10. C8×D4C2semidirect
11. C2×#102C2semidirect
15. C2×SD16C22semidirect
16. C22×C8C22semidirect
17. C4⋊Q8C22semidirect
21. C22⋊Q8C22semidirect
25. C4×Q8C22semidirect
29. C2×C4⋊C4C22semidirect
30. #106C22semidirect
31. #106C22semidirect
32. C4⋊C8C22semidirect
33. #103C22semidirect
36. #102C22semidirect
37. #102C22semidirect
38. #102C22semidirect
39. C22⋊C8C22semidirect
40. C4×C8C22semidirect
93. D4SD16semidirect

Maximal subgroups

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

subgroupindex
#5542
D422
#4862
#4812
#4762
#4652
#4602
#4502
C4×SD162
C8×D42
C2×#1022