Finite Group Databasebeta

← #3094

Subgroups — #3094

Order 128.

231 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 231 conjugacy classes of subgroups — too many to draw. Only its 94 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×#121C211yesmaximal, characteristic
2C2×#117C211yesmaximal, characteristic, complemented
3#482C214yesmaximal, complemented
4#481C212yesmaximal, complemented
5#480C212yesmaximal, complemented
6#428C212yesmaximal, complemented
7#418C212yesmaximal, complemented
8#407C211yesmaximal, characteristic, complemented
9C22×D4—21no—
10C22×D4C2211yescharacteristic
11C23×C4C2211yescharacteristic
12C2×M16C2212yes—
13#121—21no—
14#121—21no—
15#121C2212yes—
16#121C2212yes—
17#117—21no—
18#117—21no—
19#117C2212yes—
20#117C2212yescomplemented
21C2×C4⋊C4C2211yescharacteristic
22C2×C4⋊C4C2211yescharacteristic, complemented
23C2×C4⋊C4C2211yescharacteristic, complemented
24C2×C22⋊C4—21no—
25C2×C22⋊C4—21no—
26C2×C42—21no—
27#107C2214yescomplemented
28#106C2214yescomplemented
29#102C2214yescomplemented
30#102C2214yescomplemented
31C22⋊C8C2212yescomplemented
32C22⋊C8C2212yescomplemented
33C24—21no—
34C24D411yescharacteristic
35C2×D4—22no—
36C2×D4—22no—
37C2×D4—21no—
38C2×D4—24no—
39C2×D4—44no—
40C2×D4C2312yes—
41C22×C4—24no—
42C22×C4—21no—
43C22×C4—21no—
44C22×C4—21no—
45C22×C4D411yescharacteristic
46C22×C4D411yescharacteristic
47C22×C4D411yescharacteristic
48C22×C4C2311yescharacteristic
49C22×C4C2311yescharacteristic
50C22×C4C2311yescharacteristic
51M16—24no—
52C2×C8C2314yes—
53C4⋊C4—21no—
54C4⋊C4—21no—
55C4⋊C4—21no—
56C4⋊C4C2312yes—
57C4⋊C4C2312yes—
58C4⋊C4C2312yes—
59C22⋊C4—22no—
60C22⋊C4—22no—
61C22⋊C4—22no—
62C22⋊C4—22no—
63C42—22no—
64C42—22no—
65C23—22no—
66C23—22no—
67C23—22no—
68C23—41no—
69C23—42no—
70C23—42no—
71C23—42no—
72C23C2×D411yescharacteristic
73C23C2×D411yescharacteristic
74C23C2×D411yescharacteristic
75D4—24no—
76D4—44no—
77D4—44no—
78D4—42no—
79C2×C4—22no—
80C2×C4—22no—
81C2×C4—22no—
82C2×C4—22no—
83C2×C4—22no—
84C2×C4—22no—
85C2×C4—22no—
86C2×C4—22no—
87C2×C4—42no—
88C2×C4—42no—
89C2×C4—42no—
90C2×C4C2×D411yescharacteristic
91C2×C4C2×D411yescharacteristic
92C2×C4C2×D411yescharacteristic
93C2×C4#4114yes—
94C2×C4C2411yescharacteristic
95C8—24no—
96C22—21no—
97C22—21no—
98C22—21no—
99C22—21no—
100C22—21no—
101C22—21no—
102C22—21no—
103C22—44no—
104C22—42no—
105C22—42no—
106C22—42no—
107C22—84no—
108C22#12312yes—
109C22AGL(1,ℤ/8)12yes—
110C22C22×D411yescharacteristic
111C4—21no—
112C4—21no—
113C4—42no—
114C4—42no—
115C4—42no—
116C4#12312yes—
117C4C2×#4112yes—
118C2—22no—
119C2—22no—
120C2—41no—
121C2—82no—
122C2C2×#12311yescharacteristic
123C2C2×AGL(1,ℤ/8)11yescharacteristic
124C2#57511yescharacteristic
subgroupG/H
center, socleC22C22×D4
commutator, Frattini, cosocleC2×C4C24
prime radicalC2×#121C2
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
2. C2×#117C2semidirect
3. #482C2semidirect
4. #481C2semidirect
5. #480C2semidirect
6. #428C2semidirect
7. #418C2semidirect
8. #407C2semidirect
20. #117C22semidirect
22. C2×C4⋊C4C22semidirect
23. C2×C4⋊C4C22semidirect
27. #107C22semidirect
28. #106C22semidirect
29. #102C22semidirect
30. #102C22semidirect
31. C22⋊C8C22semidirect
32. C22⋊C8C22semidirect

Maximal subgroups

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

subgroupindex
C2×#1212
C2×#1172
#4822
#4812
#4802
#4282
#4182
#4072