Finite Group Databasebeta

← #7630463

Subgroups — #7630463

Order 512.

14970 conjugacy classes of subgroups; only the 3962 normal subgroups are shown.

normal subgroups only

This group has 14970 conjugacy classes of subgroups — too many to draw. Only its 3962 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
1C22×#534C216yesmaximal, complemented
2#35463C2116yesmaximal, complemented
3#35390C2116yesmaximal, complemented
4#35001C2124yesmaximal, complemented
5C2×#2273C211yesmaximal, characteristic, complemented
6C24×D4C2214yes—
7C2×#534—296no—
8C2×#534C22148yescomplemented
9C2×#534C22148yescomplemented
10C22×#123C22124yescomplemented
11C22×#121C22124yescomplemented
12C22×#120C22124yescomplemented
13C23×C22⋊C4C2213yes—
14#2856C22132yescomplemented
15#2656C22132yescomplemented
16#2539C221192yescomplemented
17#2501C22196yescomplemented
18#2433C22196yescomplemented
19#2273C22116yescomplemented
20C2×C4×C22⋊C4C22112yescomplemented
21C26C2314yes—
22C26C2311yescharacteristic
23C23×D4—296no—
24C23×D4C23124yes—
25C23×D4C23124yes—
26C23×D4C23124yescomplemented
27C24×C4C2314yes—
28C24×C4C2316yes—
29#534—2384no—
30#534—2384no—
31C2×#127C23132yescomplemented
32C2×#123—2192no—
33C2×#123C231192yescomplemented
34C2×#121—2192no—
35C2×#121C23196yes—
36C2×#121C23196yescomplemented
37C2×#120—2192no—
38C2×#120C231192yescomplemented
39C22×C4⋊C4C23124yescomplemented
40C22×C22⋊C4D4148yescomplemented
41C22×C22⋊C4C23124yes—
42C22×C22⋊C4C23112yes—
43C22×C22⋊C4C23148yescomplemented
44C22×C42C2318yescomplemented
45#399C231128yescomplemented
46#392C231128yescomplemented
47#390C231192yescomplemented
48#384C23132yescomplemented
49C4×C22⋊C4C23196yescomplemented
50C2×#95C2318yescomplemented
51C25—248no—
52C25—84no—
53C25—816no—
54C25C2×D4124yes—
55C25C24124yes—
56C25C2414yes—
57C25C2413yes—
58C22×D4—296no—
59C22×D4—296no—
60C22×D4—2192no—
61C22×D4—2192no—
62C22×D4—296no—
63C22×D4—4256no—
64C22×D4—848no—
65C22×D4C24148yes—
66C22×D4C24116yes—
67C22×D4C24196yes—
68C23×C4—248no—
69C23×C4—812no—
70C23×C4C2×D4148yescomplemented
71C23×C4C24124yes—
72C23×C4C24124yes—
73C23×C4C24112yes—
74#127—832no—
75#123—2768no—
76#121—2384no—
77#121—2384no—
78#120—2768no—
79C2×C4⋊C4—296no—
80C2×C4⋊C4C24196yes—
81C2×C22⋊C4—2192no—
82C2×C22⋊C4C2×D41192yescomplemented
83C2×C22⋊C4C2×D4196yescomplemented
84C2×C22⋊C4C241192yes—
85C2×C22⋊C4C24148yes—
86C2×C42—88no—
87C2×C42C24132yes—
88#95C24132yes—
89C24—248no—
90C24—296no—
91C24—412no—
92C24—416no—
93C24—496no—
94C24—464no—
95C24—848no—
96C24—8192no—
97C24#127164yescomplemented
98C24C22×D4148yes—
99C24C22×D4124yes—
100C24C22×D4112yes—
101C24C25116yes—
102C24C2516yes—
103C24C2511yescharacteristic
104C2×D4—2192no—
105C2×D4—296no—
106C2×D4—296no—
107C2×D4—2192no—
108C2×D4—4192no—
109C2×D4—4384no—
110C2×D4—4384no—
111C2×D4—8192no—
112C22×C4—248no—
113C22×C4—296no—
114C22×C4—424no—
115C22×C4—448no—
116C22×C4—464no—
117C22×C4—848no—
118C22×C4C22×D4196yes—
119C22×C4C22×D4148yes—
120C22×C4C25116yes—
121C22×C4C25124yes—
122C4⋊C4—2192no—
123C22⋊C4—2384no—
124C22⋊C4C22×D41192yes—
125C42—432no—
126C23—24no—
127C23—224no—
128C23—248no—
129C23—264no—
130C23—424no—
131C23—496no—
132C23—448no—
133C23—4192no—
134C23—496no—
135C23—8192no—
136C23—864no—
137C23—8256no—
138C23C2×#127164yes—
139C23C2×#127148yes—
140C23C23×D4124yes—
141C23C23×D4112yes—
142C23C23×D416yes—
143C23C2611yescharacteristic
144D4—4192no—
145D4—464no—
146D4—8192no—
147C2×C4—248no—
148C2×C4—248no—
149C2×C4—448no—
150C2×C4—496no—
151C2×C4—496no—
152C2×C4C23×D4148yes—
153C22—224no—
154C22—212no—
155C22—248no—
156C22—464no—
157C22—448no—
158C22—496no—
159C22—416no—
160C22—8192no—
161C22C22×#127124yes—
162C22C22×#12714yes—
163C22C24×D413yes—
164C22H(7,𝔽2)14yes—
165C4—248no—
166C4—416no—
167C2—224no—
168C2—24no—
169C2—816no—
170C2C23×#12711yescharacteristic
171C2#6284216yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC23C26
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ G
LECSG ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#534C2semidirect
2. #35463C2semidirect
3. #35390C2semidirect
4. #35001C2semidirect
5. C2×#2273C2semidirect
8. C2×#534C22semidirect
9. C2×#534C22semidirect
10. C22×#123C22semidirect
11. C22×#121C22semidirect
12. C22×#120C22semidirect
14. #2856C22semidirect
15. #2656C22semidirect
16. #2539C22semidirect
17. #2501C22semidirect
18. #2433C22semidirect
19. #2273C22semidirect
20. C2×C4×C22⋊C4C22semidirect
26. C23×D4C23semidirect
31. C2×#127C23semidirect
33. C2×#123C23semidirect
36. C2×#121C23semidirect
38. C2×#120C23semidirect
39. C22×C4⋊C4C23semidirect
40. C22×C22⋊C4D4semidirect
43. C22×C22⋊C4C23semidirect
44. C22×C42C23semidirect
45. #399C23semidirect
46. #392C23semidirect
47. #390C23semidirect
48. #384C23semidirect
49. C4×C22⋊C4C23semidirect
50. C2×#95C23semidirect
70. C23×C4C2×D4semidirect
82. C2×C22⋊C4C2×D4semidirect
83. C2×C22⋊C4C2×D4semidirect
97. C24#127semidirect

Maximal subgroups

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

subgroupindex
C22×#5342
#354632
#353902
#350012
C2×#22732