Finite Group Databasebeta

← #3272

Subgroups — #3272

Order 128.

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

normal subgroups only

This group has 273 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
1C2×AGL(1,ℤ/8)C212yesmaximal, complemented
2#547C211yesmaximal, characteristic, complemented
3D42C211yesmaximal, characteristic, complemented
4#492C211yesmaximal, characteristic, complemented
5#468C212yesmaximal, complemented
6#461C211yesmaximal, characteristic, complemented
7#459C211yesmaximal, characteristic, complemented
8#450C212yesmaximal, complemented
9#449C212yesmaximal, complemented
10C4×D8C211yesmaximal, characteristic, complemented
11#436C211yesmaximal, characteristic, complemented
12C2×#41C2212yescomplemented
13C22×D4—21no—
14C22×D4C2212yes—
15AGL(1,ℤ/8)—28no—
16C2×SD16C2212yes—
17C2×SD16C2212yescomplemented
18C2×SD16C2212yescomplemented
19C2×D8C2211yescharacteristic
20C2×D8C2212yescomplemented
21C2×M16C2212yescomplemented
22C4⋊Q8C2211yescharacteristic, complemented
23#127C2211yescharacteristic
24#123—22no—
25C22⋊Q8C2212yescomplemented
26#121—21no—
27#121—21no—
28#121C2212yes—
29#120—22no—
30C4×D4C2211yescharacteristic
31C4×D4C2211yescharacteristic
32C4×D4C2211yescharacteristic, complemented
33C2×C4⋊C4—21no—
34#107C2211yescharacteristic, complemented
35C4⋊C8C2211yescharacteristic, complemented
36#103C2212yescomplemented
37#102C2211yescharacteristic
38#102C2211yescharacteristic, complemented
39#102C2212yescomplemented
40C22⋊C8C2212yescomplemented
41C4×C8C2211yescharacteristic, complemented
42C24—22no—
43#41—24no—
44#41—42no—
45C2×Q8C2312yes—
46C2×D4—24no—
47C2×D4—22no—
48C2×D4—22no—
49C2×D4—22no—
50C2×D4—21no—
51C2×D4—21no—
52C2×D4—42no—
53C2×D4—44no—
54C2×D4D411yescharacteristic
55C2×D4C2312yes—
56C2×D4C2311yescharacteristic
57C2×D4C2311yescharacteristic
58C22×C4—21no—
59C22×C4—22no—
60C22×C4—21no—
61C22×C4C2312yes—
62SD16—24no—
63SD16—24no—
64SD16—42no—
65D8—24no—
66D8D414yescomplemented
67M16—24no—
68C2×C8C2311yescharacteristic
69C2×C8C2312yes—
70C2×C8C2311yescharacteristic
71C4⋊C4—21no—
72C4⋊C4—22no—
73C4⋊C4—21no—
74C4⋊C4D411yescharacteristic
75C4⋊C4C2311yescharacteristic
76C4⋊C4C2311yescharacteristic
77C22⋊C4—22no—
78C22⋊C4—21no—
79C22⋊C4—21no—
80C22⋊C4—22no—
81C22⋊C4D412yes—
82C42C2311yescharacteristic
83C23—21no—
84C23—21no—
85C23—22no—
86C23—44no—
87C23—44no—
88C23—82no—
89C23C2×D412yes—
90Q8—22no—
91Q8—22no—
92D4—22no—
93D4—22no—
94D4—21no—
95D4—21no—
96D4—22no—
97D4—44no—
98D4—44no—
99D4—44no—
100D4—42no—
101D4—81no—
102D4C2×D412yes—
103D4C2×D412yescomplemented
104C2×C4—22no—
105C2×C4—22no—
106C2×C4—21no—
107C2×C4—21no—
108C2×C4—22no—
109C2×C4—44no—
110C2×C4—42no—
111C2×C4—42no—
112C2×C4C2×D411yescharacteristic
113C2×C4C2×D412yes—
114C2×C4C2×D411yescharacteristic
115C2×C4C2411yescharacteristic
116C8—22no—
117C8—21no—
118C8C2×D412yescomplemented
119C22—22no—
120C22—22no—
121C22—22no—
122C22—22no—
123C22—22no—
124C22—41no—
125C22—41no—
126C22—41no—
127C22—41no—
128C22—42no—
129C22—42no—
130C22—42no—
131C22—88no—
132C22C22×D411yescharacteristic
133C4—22no—
134C4—21no—
135C4—42no—
136C4—41no—
137C4—41no—
138C4AGL(1,ℤ/8)12yescomplemented
139C4C22×D411yescharacteristic
140C4H(5,𝔽2)11yescharacteristic
141C2—42no—
142C2—42no—
143C2—42no—
144C2—82no—
145C2D4211yescharacteristic
146C2C2×AGL(1,ℤ/8)11yescharacteristic
147C2#57711yescharacteristic
subgroupG/H
center, socleC22C22×D4
commutator, Frattini, cosocleC2×C4C24
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. C2×AGL(1,ℤ/8)C2semidirect
2. #547C2semidirect
3. D42C2semidirect
4. #492C2semidirect
5. #468C2semidirect
6. #461C2semidirect
7. #459C2semidirect
8. #450C2semidirect
9. #449C2semidirect
10. C4×D8C2semidirect
11. #436C2semidirect
12. C2×#41C22semidirect
17. C2×SD16C22semidirect
18. C2×SD16C22semidirect
20. C2×D8C22semidirect
21. C2×M16C22semidirect
22. C4⋊Q8C22semidirect
25. C22⋊Q8C22semidirect
32. C4×D4C22semidirect
34. #107C22semidirect
35. C4⋊C8C22semidirect
36. #103C22semidirect
38. #102C22semidirect
39. #102C22semidirect
40. C22⋊C8C22semidirect
41. C4×C8C22semidirect
66. D8D4semidirect
103. D4C2×D4semidirect
118. C8C2×D4semidirect
138. C4AGL(1,ℤ/8)semidirect

Maximal subgroups

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

subgroupindex
C2×AGL(1,ℤ/8)2
#5472
D422
#4922
#4682
#4612
#4592
#4502
#4492
C4×D82
#4362