Finite Group Databasebeta

← #3484

Subgroups — #3484

Order 128.

950 conjugacy classes of subgroups; only the 432 normal subgroups are shown.

normal subgroups only

This group has 950 conjugacy classes of subgroups — too many to draw. Only its 432 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×H(5,𝔽2)C211yesmaximal, characteristic, complemented
2C23×D4C211yesmaximal, characteristic, complemented
3#546C218yesmaximal, complemented
4D42C218yesmaximal, complemented
5#535C212yesmaximal, complemented
6#534C214yesmaximal, complemented
7#525C212yesmaximal, complemented
8C2×#120C214yesmaximal, complemented
9#518C211yesmaximal, characteristic, complemented
10C25C2212yescomplemented
11H(5,𝔽2)—28no—
12C2×#41C2212yescomplemented
13C2×#41C2214yescomplemented
14C22×D4—28no—
15C22×D4C2211yescharacteristic
16C22×D4C2212yescomplemented
17C22×D4C2211yescharacteristic, complemented
18C22×D4C2218yescomplemented
19C22×D4C2214yescomplemented
20C22×D4C2214yescomplemented
21C22×D4C2218yescomplemented
22C23×C4C2211yescharacteristic, complemented
23#127C2214yescomplemented
24#124C2214yescomplemented
25#123C2214yescomplemented
26#123C2218yescomplemented
27C22⋊Q8C2214yescomplemented
28#121C2214yescomplemented
29#121C22116yescomplemented
30#121C2218yescomplemented
31#120C2214yescomplemented
32#120C2218yescomplemented
33#120C22116yescomplemented
34#120C2218yescomplemented
35C4×D4C2218yescomplemented
36C4×D4C2218yescomplemented
37#117C2212yescomplemented
38C2×C22⋊C4C2214yescomplemented
39C2×C22⋊C4C2218yescomplemented
40C24—28no—
41C24—48no—
42C24C2311yescharacteristic
43C24C2318yes—
44C24C2312yes—
45C24C2312yes—
46C24C2314yescomplemented
47#41—216no—
48#41—28no—
49C2×Q8C2312yescomplemented
50C2×D4—28no—
51C2×D4—216no—
52C2×D4—232no—
53C2×D4—216no—
54C2×D4—24no—
55C2×D4—416no—
56C2×D4D418yescomplemented
57C2×D4C2318yes—
58C2×D4C2314yes—
59C2×D4C2318yes—
60C2×D4C2314yes—
61C2×D4C2314yes—
62C2×D4C2318yescomplemented
63C2×D4C2312yescomplemented
64C2×D4C23116yescomplemented
65C22×C4—24no—
66C22×C4C2314yes—
67C22×C4C2311yescharacteristic
68C22×C4C2311yescharacteristic
69C22×C4C2314yescomplemented
70C22×C4C2314yescomplemented
71C22×C4C2318yescomplemented
72C4⋊C4C2314yescomplemented
73C4⋊C4C2318yescomplemented
74C22⋊C4C2314yes—
75C22⋊C4C2318yes—
76C22⋊C4C2318yescomplemented
77C22⋊C4C23116yescomplemented
78C22⋊C4C2318yescomplemented
79C42C2314yescomplemented
80C23—22no—
81C23—24no—
82C23—216no—
83C23—24no—
84C23—216no—
85C23—28no—
86C23—44no—
87C23—48no—
88C23—44no—
89C23—48no—
90C23—432no—
91C23C2×D418yescomplemented
92C23C2414yes—
93C23C2414yes—
94C23C2411yescharacteristic
95C23C2414yes—
96C23C2412yes—
97Q8—24no—
98D4—216no—
99D4—28no—
100D4—216no—
101D4—216no—
102D4—28no—
103D4—24no—
104D4—416no—
105D4C2×D4116yescomplemented
106C2×C4—22no—
107C2×C4—28no—
108C2×C4—28no—
109C2×C4—28no—
110C2×C4—48no—
111C2×C4C2×D414yescomplemented
112C2×C4C2418yes—
113C2×C4C2414yes—
114C2×C4C2412yes—
115C2×C4C2412yes—
116C22—28no—
117C22—22no—
118C22—28no—
119C22—22no—
120C22—22no—
121C22—24no—
122C22—24no—
123C22—28no—
124C22—28no—
125C22—48no—
126C22—432no—
127C22—48no—
128C22—88no—
129C22C22×D412yes—
130C22C22×D418yes—
131C22H(5,𝔽2)14yescomplemented
132C22C2511yescharacteristic
133C4—22no—
134C4—24no—
135C4—28no—
136C4C22×D414yes—
137C2—22no—
138C2—24no—
139C2—28no—
140C2—44no—
141C2—44no—
142C2C23×D411yescharacteristic
143C2C2×H(5,𝔽2)12yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC22C25
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ G
LECSG ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×H(5,𝔽2)C2semidirect
2. C23×D4C2semidirect
3. #546C2semidirect
4. D42C2semidirect
5. #535C2semidirect
6. #534C2semidirect
7. #525C2semidirect
8. C2×#120C2semidirect
9. #518C2semidirect
10. C25C22semidirect
12. C2×#41C22semidirect
13. C2×#41C22semidirect
16. C22×D4C22semidirect
17. C22×D4C22semidirect
18. C22×D4C22semidirect
19. C22×D4C22semidirect
20. C22×D4C22semidirect
21. C22×D4C22semidirect
22. C23×C4C22semidirect
23. #127C22semidirect
24. #124C22semidirect
25. #123C22semidirect
26. #123C22semidirect
27. C22⋊Q8C22semidirect
28. #121C22semidirect
29. #121C22semidirect
30. #121C22semidirect
31. #120C22semidirect
32. #120C22semidirect
33. #120C22semidirect
34. #120C22semidirect
35. C4×D4C22semidirect
36. C4×D4C22semidirect
37. #117C22semidirect
38. C2×C22⋊C4C22semidirect
39. C2×C22⋊C4C22semidirect
46. C24C23semidirect
49. C2×Q8C23semidirect
56. C2×D4D4semidirect
62. C2×D4C23semidirect
63. C2×D4C23semidirect
64. C2×D4C23semidirect
69. C22×C4C23semidirect
70. C22×C4C23semidirect
71. C22×C4C23semidirect
72. C4⋊C4C23semidirect
73. C4⋊C4C23semidirect
76. C22⋊C4C23semidirect
77. C22⋊C4C23semidirect
78. C22⋊C4C23semidirect
79. C42C23semidirect
91. C23C2×D4semidirect
105. D4C2×D4semidirect
111. C2×C4C2×D4semidirect
131. C22H(5,𝔽2)semidirect

Maximal subgroups

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

subgroupindex
C2×H(5,𝔽2)2
C23×D42
#5462
D422
#5352
#5342
#5252
C2×#1202
#5182