Finite Group Databasebeta

← #3294

Subgroups — #3294

Order 128.

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

normal subgroups only

This group has 247 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
1#547C211yesmaximal, characteristic, complemented
2D42C211yesmaximal, characteristic, complemented
3#486C211yesmaximal, characteristic, complemented
4#480C212yesmaximal, complemented
5#474C211yesmaximal, characteristic, complemented
6#466C212yesmaximal, complemented
7#459C211yesmaximal, characteristic, complemented
8#447C212yesmaximal, complemented
9C4×D8C211yesmaximal, characteristic, complemented
10C8×D4C211yesmaximal, characteristic, complemented
11C2×#102C212yesmaximal, complemented
12C2×#41—21no—
13C22×D4—21no—
14C22×D4C2212yes—
15C2×D8C2211yescharacteristic
16C2×D8C2212yescomplemented
17C22×C8C2212yescomplemented
18C4⋊Q8C2211yescharacteristic, complemented
19#127C2211yescharacteristic
20#123—22no—
21C22⋊Q8—21no—
22#121—21no—
23#121C2212yes—
24#121C2212yescomplemented
25#120—22no—
26C4×D4C2211yescharacteristic
27C4×D4C2211yescharacteristic
28C4×D4C2211yescharacteristic, complemented
29C2×C4⋊C4C2212yescomplemented
30#107C2211yescharacteristic, complemented
31#107C2212yescomplemented
32C4⋊C8C2211yescharacteristic, complemented
33#102C2211yescharacteristic
34#102C2212yes—
35#102C2212yescomplemented
36#102C2212yescomplemented
37#102C2211yescharacteristic, complemented
38#102C2212yescomplemented
39C22⋊C8C2212yescomplemented
40C4×C8C2211yescharacteristic, complemented
41C24—22no—
42#41—44no—
43C2×Q8—21no—
44C2×D4—24no—
45C2×D4—22no—
46C2×D4—22no—
47C2×D4—22no—
48C2×D4—21no—
49C2×D4—21no—
50C2×D4—42no—
51C2×D4—44no—
52C2×D4D411yescharacteristic
53C2×D4C2312yes—
54C2×D4C2311yescharacteristic
55C2×D4C2311yescharacteristic
56C22×C4—22no—
57C22×C4—21no—
58C22×C4—21no—
59C22×C4C2312yes—
60D8—22no—
61D8—42no—
62C2×C8—42no—
63C2×C8C2311yescharacteristic
64C2×C8C2311yescharacteristic
65C2×C8C2312yes—
66C4⋊C4—22no—
67C4⋊C4—21no—
68C4⋊C4D411yescharacteristic
69C4⋊C4C2311yescharacteristic
70C4⋊C4C2312yes—
71C4⋊C4C2311yescharacteristic
72C22⋊C4—21no—
73C22⋊C4—21no—
74C22⋊C4—22no—
75C22⋊C4—22no—
76C22⋊C4D412yes—
77C42C2311yescharacteristic
78C23—21no—
79C23—21no—
80C23—22no—
81C23—44no—
82C23—44no—
83C23—82no—
84C23C2×D412yes—
85Q8—42no—
86D4—21no—
87D4—22no—
88D4—21no—
89D4—21no—
90D4—22no—
91D4—44no—
92D4—44no—
93D4—44no—
94D4—42no—
95D4—81no—
96D4D814yescomplemented
97D4#4112yescomplemented
98C2×C4—22no—
99C2×C4—22no—
100C2×C4—21no—
101C2×C4—21no—
102C2×C4—22no—
103C2×C4—44no—
104C2×C4—42no—
105C2×C4—42no—
106C2×C4C2×D411yescharacteristic
107C2×C4C2×D412yes—
108C2×C4C2×D411yescharacteristic
109C2×C4C2411yescharacteristic
110C8—21no—
111C8—21no—
112C8—22no—
113C22—22no—
114C22—22no—
115C22—22no—
116C22—41no—
117C22—41no—
118C22—42no—
119C22—41no—
120C22—41no—
121C22—42no—
122C22—42no—
123C22—41no—
124C22—88no—
125C22C2×D814yes—
126C22C22×D411yescharacteristic
127C4—22no—
128C4—21no—
129C4—41no—
130C4—41no—
131C4—42no—
132C4C2×D812yes—
133C4C2×#4111yescharacteristic
134C4H(5,𝔽2)11yescharacteristic
135C2—24no—
136C2—42no—
137C2—81no—
138C2—82no—
139C2#54611yescharacteristic
140C2C22×D811yescharacteristic
141C2#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. #547C2semidirect
2. D42C2semidirect
3. #486C2semidirect
4. #480C2semidirect
5. #474C2semidirect
6. #466C2semidirect
7. #459C2semidirect
8. #447C2semidirect
9. C4×D8C2semidirect
10. C8×D4C2semidirect
11. C2×#102C2semidirect
16. C2×D8C22semidirect
17. C22×C8C22semidirect
18. C4⋊Q8C22semidirect
24. #121C22semidirect
28. C4×D4C22semidirect
29. C2×C4⋊C4C22semidirect
30. #107C22semidirect
31. #107C22semidirect
32. C4⋊C8C22semidirect
35. #102C22semidirect
36. #102C22semidirect
37. #102C22semidirect
38. #102C22semidirect
39. C22⋊C8C22semidirect
40. C4×C8C22semidirect
96. D4D8semidirect
97. D4#41semidirect

Maximal subgroups

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

subgroupindex
#5472
D422
#4862
#4802
#4742
#4662
#4592
#4472
C4×D82
C8×D42
C2×#1022