Finite Group Databasebeta

← #3194

Subgroups — #3194

Order 128.

233 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 233 conjugacy classes of subgroups — too many to draw. Only its 86 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#525C211yesmaximal, characteristic, complemented
2C2×#121C211yesmaximal, characteristic, complemented
3#483C211yesmaximal, characteristic, complemented
4#482C212yesmaximal, complemented
5#480C211yesmaximal, characteristic, complemented
6#469C212yesmaximal, complemented
7#468C212yesmaximal, complemented
8#450C211yesmaximal, characteristic, complemented
9#449C212yesmaximal, complemented
10#447C211yesmaximal, characteristic, complemented
11#407C211yesmaximal, characteristic, complemented
12C2×#41C2211yescharacteristic, complemented
13C22×D4—21no—
14C22×D4C2211yescharacteristic
15C23×C4C2211yescharacteristic
16C2×SD16C2212yescomplemented
17C2×D8C2212yescomplemented
18C2×M16C2212yescomplemented
19#123—21no—
20C22⋊Q8C2211yescharacteristic
21C22⋊Q8C2211yescharacteristic, complemented
22#121—21no—
23#121—21no—
24#121C2212yes—
25#121C2212yes—
26#121C2211yescharacteristic
27#121C2211yescharacteristic, complemented
28#120—21no—
29C4×D4—21no—
30C4×D4—21no—
31#117C2211yescharacteristic, complemented
32C2×C4⋊C4C2211yescharacteristic
33C2×C22⋊C4—21no—
34#107C2212yescomplemented
35#106C2212yescomplemented
36#103C2212yescomplemented
37#102C2212yescomplemented
38#102C2212yescomplemented
39#102C2212yescomplemented
40C22⋊C8C2211yescharacteristic
41C22⋊C8C2211yescharacteristic
42C22⋊C8C2212yescomplemented
43C24—21no—
44C24D411yescharacteristic
45#41—42no—
46C2×Q8C2311yescharacteristic
47C2×D4—21no—
48C2×D4—21no—
49C2×D4—21no—
50C2×D4—22no—
51C2×D4—22no—
52C2×D4—21no—
53C2×D4—42no—
54C2×D4—44no—
55C2×D4C2312yes—
56C2×D4C2311yescharacteristic
57C22×C4—21no—
58C22×C4—21no—
59C22×C4—42no—
60C22×C4D411yescharacteristic
61C22×C4D411yescharacteristic
62C22×C4D411yescharacteristic
63C22×C4C2311yescharacteristic
64C22×C4C2311yescharacteristic
65C22×C4C2311yescharacteristic
66SD16—42no—
67D8—42no—
68M16—42no—
69C2×C8C2312yes—
70C2×C8C2312yes—
71C4⋊C4—21no—
72C4⋊C4—21no—
73C4⋊C4—21no—
74C4⋊C4C2311yescharacteristic
75C4⋊C4C2311yescharacteristic
76C4⋊C4C2312yes—
77C22⋊C4—21no—
78C22⋊C4—21no—
79C22⋊C4—21no—
80C22⋊C4—22no—
81C22⋊C4—22no—
82C22⋊C4—21no—
83C22⋊C4—21no—
84C42—21no—
85C23—22no—
86C23—21no—
87C23—22no—
88C23—41no—
89C23—41no—
90C23—42no—
91C23—42no—
92C23—42no—
93C23C2×D411yescharacteristic
94C23C2×D411yescharacteristic
95C23C2×D411yescharacteristic
96Q8—22no—
97D4—22no—
98D4—22no—
99D4—22no—
100D4—42no—
101D4—42no—
102D4—44no—
103D4—44no—
104D4—42no—
105D4—42no—
106C2×C4—22no—
107C2×C4—22no—
108C2×C4—22no—
109C2×C4—21no—
110C2×C4—21no—
111C2×C4—21no—
112C2×C4—22no—
113C2×C4—41no—
114C2×C4—41no—
115C2×C4—41no—
116C2×C4—42no—
117C2×C4—42no—
118C2×C4C2×D411yescharacteristic
119C2×C4C2×D411yescharacteristic
120C2×C4C2×D411yescharacteristic
121C2×C4C2411yescharacteristic
122C8—22no—
123C8—22no—
124C22—21no—
125C22—21no—
126C22—21no—
127C22—21no—
128C22—21no—
129C22—21no—
130C22—21no—
131C22—21no—
132C22—44no—
133C22—41no—
134C22—42no—
135C22—42no—
136C22—41no—
137C22—41no—
138C22—42no—
139C22—84no—
140C22AGL(1,ℤ/8)12yes—
141C22C22×D411yescharacteristic
142C4—21no—
143C4—21no—
144C4—21no—
145C4—41no—
146C4—42no—
147C4—41no—
148C4—41no—
149C4H(5,𝔽2)12yes—
150C2—22no—
151C2—41no—
152C2—41no—
153C2—81no—
154C2—82no—
155C2#53411yescharacteristic
156C2C2×AGL(1,ℤ/8)11yescharacteristic
157C2#57511yescharacteristic
subgroupG/H
center, socleC22C22×D4
commutator, Frattini, cosocleC2×C4C24
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
1. #525C2semidirect
2. C2×#121C2semidirect
3. #483C2semidirect
4. #482C2semidirect
5. #480C2semidirect
6. #469C2semidirect
7. #468C2semidirect
8. #450C2semidirect
9. #449C2semidirect
10. #447C2semidirect
11. #407C2semidirect
12. C2×#41C22semidirect
16. C2×SD16C22semidirect
17. C2×D8C22semidirect
18. C2×M16C22semidirect
21. C22⋊Q8C22semidirect
27. #121C22semidirect
31. #117C22semidirect
34. #107C22semidirect
35. #106C22semidirect
36. #103C22semidirect
37. #102C22semidirect
38. #102C22semidirect
39. #102C22semidirect
42. C22⋊C8C22semidirect

Maximal subgroups

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

subgroupindex
#5252
C2×#1212
#4832
#4822
#4802
#4692
#4682
#4502
#4492
#4472
#4072