Finite Group Databasebeta

← #3200

Subgroups — #3200

Order 128.

205 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 205 conjugacy classes of subgroups — too many to draw. Only its 84 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#536C211yesmaximal, characteristic, complemented
2#535C211yesmaximal, characteristic
3#485C212yesmaximal, complemented
4#482C212yesmaximal, complemented
5#468C212yesmaximal, complemented
6#465C212yesmaximal, complemented
7#450C212yesmaximal, complemented
8#448C212yesmaximal, complemented
9#408C211yesmaximal, characteristic, complemented
10C2×#41C2211yescharacteristic
11C22×Q8C2211yescharacteristic, complemented
12C22×D4C2211yescharacteristic
13C2×SD16C2212yes—
14C2×SD16C2212yes—
15C2×M16C2211yescharacteristic, complemented
16C22×C8C2211yescharacteristic, complemented
17C4⋊Q8—21no—
18#127—21no—
19#124—21no—
20#124—21no—
21#123—21no—
22#123—21no—
23C22⋊Q8C2212yes—
24C22⋊Q8C2212yescomplemented
25#121C2212yes—
26#121C2212yes—
27#120—21no—
28#120—21no—
29#117C2211yescharacteristic
30#117C2211yescharacteristic, complemented
31#107C2212yescomplemented
32#106C2212yescomplemented
33#103C2212yescomplemented
34#103C2212yescomplemented
35#102C2212yes—
36#102C2212yes—
37C22⋊C8C2212yes—
38C22⋊C8C2212yescomplemented
39C24—21no—
40#41—41no—
41#41—41no—
42C2×Q8—21no—
43C2×Q8—41no—
44C2×Q8—41no—
45C2×Q8D411yescharacteristic
46C2×Q8C2312yes—
47C2×D4—22no—
48C2×D4—22no—
49C2×D4—21no—
50C2×D4—41no—
51C2×D4—41no—
52C2×D4D411yescharacteristic
53C2×D4D411yescharacteristic
54C2×D4D411yescharacteristic
55C2×D4C2312yes—
56C22×C4—21no—
57C22×C4C2311yescharacteristic
58C22×C4C2311yescharacteristic
59C22×C4C2311yescharacteristic
60SD16—42no—
61SD16—42no—
62M16—41no—
63C2×C8—41no—
64C2×C8C2312yes—
65C2×C8C2312yes—
66C4⋊C4—22no—
67C4⋊C4—22no—
68C4⋊C4C2312yes—
69C4⋊C4C2312yes—
70C22⋊C4—21no—
71C22⋊C4—21no—
72C22⋊C4—22no—
73C22⋊C4—22no—
74C22⋊C4—22no—
75C22⋊C4—22no—
76C42—21no—
77C42—21no—
78C23—22no—
79C23—42no—
80C23—82no—
81C23C2×D411yescharacteristic
82C23C2×D411yescharacteristic
83C23C2×D411yescharacteristic
84Q8—22no—
85Q8—22no—
86Q8—41no—
87Q8—42no—
88D4—22no—
89D4—22no—
90D4—41no—
91D4—41no—
92D4—41no—
93D4—42no—
94D4—82no—
95D4—82no—
96C2×C4—21no—
97C2×C4—21no—
98C2×C4—21no—
99C2×C4—21no—
100C2×C4—21no—
101C2×C4—21no—
102C2×C4—22no—
103C2×C4—22no—
104C2×C4—22no—
105C2×C4—42no—
106C2×C4C2×D411yescharacteristic
107C2×C4C2×D411yescharacteristic
108C2×C4C2×D411yescharacteristic
109C2×C4C2411yescharacteristic
110C8—22no—
111C8—22no—
112C22—21no—
113C22—21no—
114C22—21no—
115C22—22no—
116C22—22no—
117C22—22no—
118C22—42no—
119C22—42no—
120C22—42no—
121C22—84no—
122C22C22×D411yescharacteristic
123C4—21no—
124C4—21no—
125C4—21no—
126C4—42no—
127C4—42no—
128C4—42no—
129C4H(5,𝔽2)11yescharacteristic
130C4H(5,𝔽2)11yescharacteristic
131C2—41no—
132C2—41no—
133C2—41no—
134C2—82no—
135C2#53411yescharacteristic
136C2#57712yes—
subgroupG/H
center, socleC22C22×D4
commutator, Frattini, cosocleC2×C4C24
prime radical#535C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×#41 ◁ G
LECSG ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #536C2semidirect
3. #485C2semidirect
4. #482C2semidirect
5. #468C2semidirect
6. #465C2semidirect
7. #450C2semidirect
8. #448C2semidirect
9. #408C2semidirect
11. C22×Q8C22semidirect
15. C2×M16C22semidirect
16. C22×C8C22semidirect
24. C22⋊Q8C22semidirect
30. #117C22semidirect
31. #107C22semidirect
32. #106C22semidirect
33. #103C22semidirect
34. #103C22semidirect
38. C22⋊C8C22semidirect

Maximal subgroups

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

subgroupindex
#5362
#5352
#4852
#4822
#4682
#4652
#4502
#4482
#4082