Finite Group Databasebeta

← #7658233

Subgroups — #7658233

Order 512.

4822 conjugacy classes of subgroups.

The subgroup lattice has 333 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#40627C2124yesmaximal, complemented
2#40610C2112yesmaximal, complemented
3#37480C216yesmaximal, complemented
4#37475C213yesmaximal
5#36674C212yesmaximal, complemented
6C2×#2840C211yesmaximal, characteristic
7#35705C2112yesmaximal, complemented
8#34176C213yesmaximal
9C23×C22⋊C4C2211yescharacteristic
10#2842C2216yes—
11#2842C2213yes—
12#2841C22124yescomplemented
13#2840C2211yescharacteristic
14#2840C2213yes—
15#2840C22112yes—
16#2840C2213yes—
17#2788C22124yescomplemented
18#2763C22112yescomplemented
19#2763C22112yescomplemented
20#2742C22112yescomplemented
21#2729C22124yescomplemented
22#2677C22112yescomplemented
23#2665C22124yescomplemented
24#2665C22124yescomplemented
25#2628C22124yescomplemented
26#2626C22112yes—
27#2626C22124yes—
28#2607C22124yescomplemented
29#2589C22124yescomplemented
30#2574C22124yescomplemented
31#2549C22112yescomplemented
32#2505C22124yescomplemented
33#2492C22124yescomplemented
34#2465C22124yescomplemented
35#2439C22124yescomplemented
36#2439C22124yescomplemented
37#2437C22124yes—
38#2437C22112yes—
39#2434C22112yescomplemented
40#2396C22124yescomplemented
41C2×#393C22112yes—
42#2368C22112yescomplemented
43#2342C22112yes—
44#2342C22112yes—
45#2342C22112yes—
46#2342C22112yes—
47Q8×C22⋊C4C2214yescomplemented
48#2315C2216yes—
49#2312C22112yes—
50#2280C2216yes—
51#2280C2216yes—
52C2×#379C2213yes—
53C2×#379C2213yes—
54#2273C2216yes—
55C26C2311yescharacteristic
56C23×Q8C2314yes—
57C24×C4C2312yes—
58C2×C4⋊Q8C23124yescomplemented
59C2×C22⋊Q8C23112yes—
60C2×C22⋊Q8C23124yes—
61C2×C4×Q8C2318yescomplemented
62C2×#117C23112yes—
63C2×#117C23112yes—
64C22×C22⋊C4Q8116yescomplemented
65C22×C22⋊C4C2316yes—
66C22×C22⋊C4C2316yes—
67C22×C22⋊C4C23112yes—
68C22×C22⋊C4C23112yes—
69C22×C22⋊C4C2313yes—
70C22×C22⋊C4C2313yes—
71C22×C22⋊C4C2316yes—
72#400C23124yescomplemented
73#400C23124yescomplemented
74#400C23124yescomplemented
75#399C23124yes—
76#399C23112yes—
77#399C23112yes—
78#398C23124yescomplemented
79#398C23124yescomplemented
80#396C23124yes—
81#396C23124yes—
82#396C23124yes—
83#396C23124yes—
84#395C23124yescomplemented
85#395C23124yescomplemented
86#395C23124yescomplemented
87#393C23112yes—
88#393C23112yes—
89#393C23124yes—
90#393C23124yes—
91#393C23112yes—
92#393C23112yes—
93#393C23124yes—
94#393C23112yes—
95#393C23112yes—
96#391C23124yescomplemented
97#391C23124yescomplemented
98#391C23124yescomplemented
99#391C23124yescomplemented
100#389C23124yescomplemented
101#389C23124yescomplemented
102#388C23124yes—
103#388C23124yes—
104#388C23124yes—
105#388C23124yes—
106#387C23124yescomplemented
107#387C23124yescomplemented
108#387C23124yescomplemented
109#387C23124yescomplemented
110#385C23112yes—
111#385C23124yes—
112#385C23112yes—
113#385C23124yes—
114#385C23124yes—
115#385C23124yes—
116#382C23124yescomplemented
117#381C23112yes—
118#381C23112yes—
119#380C23112yes—
120#380C23112yes—
121#380C23124yes—
122#380C23112yes—
123#380C23112yes—
124#380C23112yes—
125#379C2316yes—
126#379C2316yes—
127#379C2313yes—
128#379C2313yes—
129#379C23112yes—
130#379C2316yes—
131#379C23112yes—
132C4×C4⋊C4C23124yescomplemented
133C4×C22⋊C4C23112yes—
134C4×C22⋊C4C23112yes—
135C4×C22⋊C4C23124yes—
136C4×C22⋊C4C23112yes—
137C4×C22⋊C4C23112yes—
138C4×C22⋊C4C23112yes—
139C4×C22⋊C4C23112yes—
140C2×#95C23112yes—
141C25—44no—
142C25—84no—
143C25C2×Q818yes—
144C25C2413yes—
145C25C2413yes—
146C25C2411yescharacteristic
147C22×Q8—48no—
148C22×Q8—88no—
149C22×Q8C2414yes—
150C22×Q8C24112yes—
151C22×Q8C24112yes—
152C23×C4—84no—
153C23×C4C2×Q8116yescomplemented
154C23×C4C24112yes—
155C23×C4C24112yes—
156C23×C4C2416yes—
157C23×C4C2416yes—
158C4⋊Q8—824no—
159C22⋊Q8—424no—
160C22⋊Q8—824no—
161C4×Q8—88no—
162#117—812no—
163#117—812no—
164C2×C4⋊C4C24124yes—
165C2×C4⋊C4C24124yes—
166C2×C4⋊C4C24124yes—
167C2×C4⋊C4C24124yes—
168C2×C4⋊C4C24124yes—
169C2×C4⋊C4C24124yes—
170C2×C4⋊C4C24112yes—
171C2×C22⋊C4—412no—
172C2×C22⋊C4—412no—
173C2×C22⋊C4—812no—
174C2×C22⋊C4—812no—
175C2×C22⋊C4C2×Q8124yes—
176C2×C22⋊C4C2×Q8124yes—
177C2×C22⋊C4C2×Q8148yescomplemented
178C2×C22⋊C4C24112yes—
179C2×C22⋊C4C24112yes—
180C2×C22⋊C4C24112yes—
181C2×C22⋊C4C24112yes—
182C2×C22⋊C4C24112yes—
183C2×C22⋊C4C24124yes—
184C2×C22⋊C4C24124yes—
185C2×C22⋊C4C2416yes—
186C2×C22⋊C4C2416yes—
187C2×C22⋊C4C24112yes—
188C2×C22⋊C4C24112yes—
189C2×C22⋊C4C2411yescharacteristic
190C2×C22⋊C4C2413yes—
191C2×C22⋊C4C2416yes—
192C2×C22⋊C4C2413yes—
193C2×C22⋊C4C2413yes—
194C2×C42C24124yes—
195C2×C42C24124yes—
196#95C24112yes—
197#95C24112yes—
198#95C24124yes—
199#95C24124yes—
200#95C24124yes—
201#95C24124yes—
202#95C24124yes—
203#95C24124yes—
204#95C24112yes—
205#95C24112yes—
206#95C24124yes—
207C24—212no—
208C24—44no—
209C24—42no—
210C24—412no—
211C24—412no—
212C24—416no—
213C24—46no—
214C24—816no—
215C24—832no—
216C24C22×Q8112yes—
217C24C22×Q8112yes—
218C24C22×Q814yes—
219C24C2513yes—
220C24C2513yes—
221C24C2511yescharacteristic
222C2×Q8—48no—
223C2×Q8—44no—
224C2×Q8—424no—
225C2×Q8—412no—
226C2×Q8—412no—
227C2×Q8—424no—
228C2×Q8—832no—
229C22×C4—224no—
230C22×C4—412no—
231C22×C4—412no—
232C22×C4—424no—
233C22×C4—412no—
234C22×C4—424no—
235C22×C4—816no—
236C22×C4C22×Q8124yes—
237C22×C4C22×Q8124yes—
238C22×C4C25112yes—
239C22×C4C25112yes—
240C22×C4C25124yes—
241C22×C4C2512yes—
242C22×C4C2516yes—
243C4⋊C4—424no—
244C4⋊C4—424no—
245C4⋊C4—424no—
246C4⋊C4—424no—
247C4⋊C4—412no—
248C4⋊C4—424no—
249C4⋊C4—424no—
250C22⋊C4—224no—
251C22⋊C4—412no—
252C22⋊C4—46no—
253C22⋊C4—412no—
254C22⋊C4—46no—
255C22⋊C4—412no—
256C22⋊C4—412no—
257C22⋊C4—412no—
258C22⋊C4—412no—
259C22⋊C4—424no—
260C22⋊C4—424no—
261C22⋊C4C22×Q814yes—
262C22⋊C4C22×Q8112yes—
263C22⋊C4C22×Q8124yes—
264C22⋊C4C22×Q8112yes—
265C22⋊C4C22×Q8112yes—
266C42—424no—
267C42—424no—
268C23—24no—
269C23—26no—
270C23—26no—
271C23—212no—
272C23—212no—
273C23—224no—
274C23—48no—
275C23—44no—
276C23—412no—
277C23—424no—
278C23—424no—
279C23—416no—
280C23—824no—
281C23—824no—
282C23—864no—
283C23C23×Q816yes—
284C23C23×Q816yes—
285C23C23×Q812yes—
286C23#58514yes—
287C23C2611yescharacteristic
288Q8—416no—
289Q8—824no—
290Q8—1612no—
291C2×C4—24no—
292C2×C4—212no—
293C2×C4—224no—
294C2×C4—212no—
295C2×C4—212no—
296C2×C4—224no—
297C2×C4—224no—
298C2×C4—224no—
299C2×C4—224no—
300C2×C4—424no—
301C2×C4—424no—
302C2×C4—824no—
303C2×C4—1612no—
304C2×C4C23×Q814yes—
305C2×C4C23×Q8112yes—
306C22—28no—
307C22—24no—
308C22—212no—
309C22—212no—
310C22—26no—
311C22—26no—
312C22—416no—
313C22—412no—
314C22—412no—
315C22—824no—
316C22—824no—
317C22#345814yes—
318C22C24×Q811yescharacteristic
319C22C2×#58512yes—
320C22H(7,𝔽2)14yes—
321C4—24no—
322C4—212no—
323C4—412no—
324C4—412no—
325C4—424no—
326C2—24no—
327C2—46no—
328C2—46no—
329C2C2×#345811yescharacteristic
330C2#6284312yes—
331C2#6304614yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC23C26
prime radicalC26C23
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ G
LECSG ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #40627C2semidirect
2. #40610C2semidirect
3. #37480C2semidirect
5. #36674C2semidirect
7. #35705C2semidirect
12. #2841C22semidirect
17. #2788C22semidirect
18. #2763C22semidirect
19. #2763C22semidirect
20. #2742C22semidirect
21. #2729C22semidirect
22. #2677C22semidirect
23. #2665C22semidirect
24. #2665C22semidirect
25. #2628C22semidirect
28. #2607C22semidirect
29. #2589C22semidirect
30. #2574C22semidirect
31. #2549C22semidirect
32. #2505C22semidirect
33. #2492C22semidirect
34. #2465C22semidirect
35. #2439C22semidirect
36. #2439C22semidirect
39. #2434C22semidirect
40. #2396C22semidirect
42. #2368C22semidirect
47. Q8×C22⋊C4C22semidirect
58. C2×C4⋊Q8C23semidirect
61. C2×C4×Q8C23semidirect
64. C22×C22⋊C4Q8semidirect
72. #400C23semidirect
73. #400C23semidirect
74. #400C23semidirect
78. #398C23semidirect
79. #398C23semidirect
84. #395C23semidirect
85. #395C23semidirect
86. #395C23semidirect
96. #391C23semidirect
97. #391C23semidirect
98. #391C23semidirect
99. #391C23semidirect
100. #389C23semidirect
101. #389C23semidirect
106. #387C23semidirect
107. #387C23semidirect
108. #387C23semidirect
109. #387C23semidirect
116. #382C23semidirect
132. C4×C4⋊C4C23semidirect
153. C23×C4C2×Q8semidirect
177. C2×C22⋊C4C2×Q8semidirect

Maximal subgroups

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

subgroupindex
#406272
#406102
#374802
#374752
#366742
C2×#28402
#357052
#341762