Finite Group Databasebeta

← #3114

Subgroups — #3114

Order 128.

225 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 225 conjugacy classes of subgroups — too many to draw. Only its 90 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#529C211yesmaximal, characteristic, complemented
2C2×C4×D4C211yesmaximal, characteristic, complemented
3#483C211yesmaximal, characteristic, complemented
4#480C211yesmaximal, characteristic, complemented
5#478C212yesmaximal, complemented
6#463C212yesmaximal, complemented
7#450C211yesmaximal, characteristic, complemented
8#447C211yesmaximal, characteristic
9#442C212yesmaximal, complemented
10#440C212yesmaximal, complemented
11#432C211yesmaximal, characteristic, complemented
12C22×D4C2211yescharacteristic
13C23×C4—21no—
14C2×SD16C2212yes—
15C2×D8C2212yes—
16#126—21no—
17#125C2211yescharacteristic, complemented
18#124C2211yescharacteristic, complemented
19#123—21no—
20C22⋊Q8C2211yescharacteristic
21#121C2211yescharacteristic
22C4×Q8C2211yescharacteristic, complemented
23C4×D4—21no—
24C4×D4—21no—
25C4×D4—21no—
26C4×D4C2212yes—
27C4×D4C2212yes—
28C4×D4C2211yescharacteristic, complemented
29#117—21no—
30C2×C4⋊C4C2211yescharacteristic
31C2×C22⋊C4—21no—
32C2×C42C2211yescharacteristic
33#107C2212yescomplemented
34#106C2212yescomplemented
35C4⋊C8C2212yescomplemented
36#103C2212yescomplemented
37#102C2212yes—
38#102C2212yes—
39#102C2212yescomplemented
40C22⋊C8C2211yescharacteristic
41C22⋊C8C2211yescharacteristic
42#97C2212yescomplemented
43C24—21no—
44C2×Q8C2311yescharacteristic
45C2×D4—24no—
46C2×D4—21no—
47C2×D4—21no—
48C2×D4C2311yescharacteristic
49C2×D4C2312yes—
50C22×C4—21no—
51C22×C4—21no—
52C22×C4—22no—
53C22×C4—22no—
54C22×C4—44no—
55C22×C4D411yescharacteristic
56C22×C4D411yescharacteristic
57C22×C4C2311yescharacteristic
58SD16—24no—
59D8—24no—
60C2×C8C2312yes—
61C2×C8C2312yes—
62C4⋊C4—21no—
63C4⋊C4—21no—
64C4⋊C4—21no—
65C4⋊C4—21no—
66C4⋊C4—21no—
67C4⋊C4C2311yescharacteristic
68C4⋊C4C2311yescharacteristic
69C4⋊C4C2312yes—
70C22⋊C4—21no—
71C22⋊C4—22no—
72C22⋊C4—22no—
73C22⋊C4—21no—
74C22⋊C4—21no—
75C22⋊C4—21no—
76C22⋊C4—21no—
77C42—21no—
78C42D411yescharacteristic
79C42D411yescharacteristic
80C42C2311yescharacteristic
81C42C2311yescharacteristic
82C23—24no—
83C23—22no—
84C23—21no—
85C23—42no—
86C23—42no—
87C23C2×D411yescharacteristic
88Q8—22no—
89D4—24no—
90D4—22no—
91D4—22no—
92D4—42no—
93D4#4114yes—
94C2×C4—22no—
95C2×C4—22no—
96C2×C4—21no—
97C2×C4—21no—
98C2×C4—21no—
99C2×C4—22no—
100C2×C4—41no—
101C2×C4—42no—
102C2×C4—44no—
103C2×C4—44no—
104C2×C4—42no—
105C2×C4C2×D411yescharacteristic
106C2×C4C2×D411yescharacteristic
107C2×C4C2×D411yescharacteristic
108C2×C4C2×D411yescharacteristic
109C2×C4C2×D411yescharacteristic
110C2×C4C2411yescharacteristic
111C8—22no—
112C8—22no—
113C22—21no—
114C22—21no—
115C22—24no—
116C22—48no—
117C22—42no—
118C22—42no—
119C22—41no—
120C22—41no—
121C22—41no—
122C22AGL(1,ℤ/8)12yes—
123C22C22×D411yescharacteristic
124C4—21no—
125C4—21no—
126C4—21no—
127C4—21no—
128C4—21no—
129C4—41no—
130C4—42no—
131C4—41no—
132C4—41no—
133C4C2×#4112yes—
134C2—22no—
135C2—44no—
136C2—81no—
137C2#52511yescharacteristic
138C2C2×AGL(1,ℤ/8)11yescharacteristic
139C2#57511yescharacteristic
subgroupG/H
center, socleC22C22×D4
commutator, Frattini, cosocleC2×C4C24
prime radical#447C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #529C2semidirect
2. C2×C4×D4C2semidirect
3. #483C2semidirect
4. #480C2semidirect
5. #478C2semidirect
6. #463C2semidirect
7. #450C2semidirect
9. #442C2semidirect
10. #440C2semidirect
11. #432C2semidirect
17. #125C22semidirect
18. #124C22semidirect
22. C4×Q8C22semidirect
28. C4×D4C22semidirect
33. #107C22semidirect
34. #106C22semidirect
35. C4⋊C8C22semidirect
36. #103C22semidirect
39. #102C22semidirect
42. #97C22semidirect

Maximal subgroups

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

subgroupindex
#5292
C2×C4×D42
#4832
#4802
#4782
#4632
#4502
#4472
#4422
#4402
#4322