Finite Group Databasebeta

← #7805056

Subgroups — #7805056

Order 512.

4950 conjugacy classes of subgroups.

The subgroup lattice has 319 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#59196C2124yesmaximal, complemented
2#47196C212yesmaximal, complemented
3#46838C216yesmaximal, complemented
4#41426C2112yesmaximal, complemented
5#37475C211yesmaximal, characteristic
6#36013C2112yesmaximal, complemented
7C2×#2483C213yesmaximal
8#34176C213yesmaximal
9C23×C22⋊C4C2213yes—
10#2842C2213yes—
11#2840C2211yescharacteristic
12#2835C2214yescomplemented
13#2834C2214yescomplemented
14#2745C22124yescomplemented
15#2744C22124yescomplemented
16#2739C22124yescomplemented
17#2738C22124yescomplemented
18#2652C22124yescomplemented
19#2626C22112yes—
20#2575C22124yescomplemented
21#2573C22124yescomplemented
22#2569C22112yescomplemented
23#2527C22124yescomplemented
24#2523C22112yescomplemented
25#2517C22112yescomplemented
26#2515C22124yescomplemented
27#2505C22124yescomplemented
28#2503C22112yescomplemented
29#2493C22124yescomplemented
30#2493C22124yescomplemented
31#2484C22124yescomplemented
32#2483C22112yes—
33#2483C2216yes—
34#2483C22112yes—
35#2483C22112yes—
36#2483C2213yes—
37#2483C2213yes—
38#2438C22124yescomplemented
39#2437C22112yes—
40#2401C2218yescomplemented
41C2×#397C22112yes—
42#2368C22112yescomplemented
43#2344C22112yescomplemented
44#2343C22124yescomplemented
45#2342C22124yes—
46#2342C22124yes—
47#2315C2216yes—
48#2312C2216yes—
49#2312C2216yes—
50C2×#385C22112yes—
51#2280C22112yes—
52C2×#381C22112yes—
53C2×#379C2213yes—
54C2×#379C2211yescharacteristic
55#2273C2216yes—
56C26C2311yescharacteristic
57C24×C4C2316yes—
58C2×C22⋊Q8C23112yes—
59C2×#117C23124yes—
60C22×C4⋊C4C23112yes—
61C22×C22⋊C4—224no—
62C22×C22⋊C4C2316yes—
63C22×C22⋊C4C2313yes—
64C22×C22⋊C4C23112yes—
65C22×C22⋊C4C23112yes—
66C22×C22⋊C4C2313yes—
67C22×C22⋊C4C2316yes—
68C22×C22⋊C4C2316yes—
69C22×C22⋊C4C23112yes—
70#401C2318yescomplemented
71#401C2318yescomplemented
72#400C23124yescomplemented
73#400C23124yescomplemented
74#400C23124yescomplemented
75#400C23124yescomplemented
76#400C23124yescomplemented
77#399C23112yes—
78#399C2314yes—
79#398C23124yescomplemented
80#398C23124yescomplemented
81#397C23112yes—
82#397C23124yes—
83#397C23112yes—
84#397C23124yes—
85#397C23112yes—
86#397C23112yes—
87#396C23124yes—
88#396C2318yes—
89#395C2318yescomplemented
90#393C2314yes—
91#393C23112yes—
92#391C23124yescomplemented
93#389C23124yescomplemented
94#388C23124yes—
95#388C23124yes—
96#388C23124yes—
97#388C23124yes—
98#387C23124yescomplemented
99#387C23124yescomplemented
100#387C23124yescomplemented
101#387C23124yescomplemented
102#387C23124yescomplemented
103#387C23124yescomplemented
104#387C23124yescomplemented
105#387C23124yescomplemented
106#385C23124yes—
107#385C23112yes—
108#385C23112yes—
109#385C23124yes—
110#385C23112yes—
111#385C23124yes—
112#385C23124yes—
113#385C23112yes—
114#385C23112yes—
115#385C23112yes—
116#383C23124yescomplemented
117#382C23124yescomplemented
118#381C23112yes—
119#381C23112yes—
120#381C23124yes—
121#381C23124yes—
122#381C23124yes—
123#381C23112yes—
124#381C23112yes—
125#380C23112yes—
126#380C23112yes—
127#380C23124yes—
128#380C23112yes—
129#379C2312yes—
130#379C2313yes—
131#379C2313yes—
132#379C2316yes—
133#379C23112yes—
134#379C2316yes—
135C4×C4⋊C4C23124yescomplemented
136C4×C22⋊C4C23112yes—
137C4×C22⋊C4C23112yes—
138C4×C22⋊C4C23112yes—
139C4×C22⋊C4C23124yes—
140C4×C22⋊C4C23112yes—
141C4×C22⋊C4C23124yes—
142#376C2318yescomplemented
143C2×#95C23112yes—
144C2×#95C23124yes—
145C25—412no—
146C25#4118yes—
147C25C2413yes—
148C25C2413yes—
149C25C2411yescharacteristic
150C22×Q8C2414yes—
151C23×C4—224no—
152C23×C4—424no—
153C23×C4C24112yes—
154C23×C4C24112yes—
155C23×C4C24112yes—
156C23×C4C2416yes—
157C23×C4C2416yes—
158C23×C4C2414yes—
159C22⋊Q8—424no—
160#117—448no—
161C2×C4⋊C4—424no—
162C2×C4⋊C4C24112yes—
163C2×C4⋊C4C24124yes—
164C2×C4⋊C4C24112yes—
165C2×C4⋊C4C24124yes—
166C2×C4⋊C4C24124yes—
167C2×C4⋊C4C24124yes—
168C2×C4⋊C4C24112yes—
169C2×C22⋊C4—248no—
170C2×C22⋊C4—248no—
171C2×C22⋊C4—212no—
172C2×C22⋊C4—212no—
173C2×C22⋊C4—224no—
174C2×C22⋊C4—412no—
175C2×C22⋊C4—424no—
176C2×C22⋊C4—412no—
177C2×C22⋊C4C2416yes—
178C2×C22⋊C4C2416yes—
179C2×C22⋊C4C2416yes—
180C2×C22⋊C4C24112yes—
181C2×C22⋊C4C24112yes—
182C2×C22⋊C4C2416yes—
183C2×C22⋊C4C24112yes—
184C2×C22⋊C4C24124yes—
185C2×C22⋊C4C24112yes—
186C2×C22⋊C4C2416yes—
187C2×C22⋊C4C2416yes—
188C2×C22⋊C4C24112yes—
189C2×C22⋊C4C24112yes—
190C2×C22⋊C4C24112yes—
191C2×C42C24124yes—
192C2×C42C24124yes—
193#95C24124yes—
194#95C24112yes—
195#95C24124yes—
196#95C24112yes—
197#95C24124yes—
198#95C24124yes—
199#95C24124yes—
200#95C2418yes—
201#95C24124yes—
202#95C24124yes—
203#95C24124yes—
204#95C24124yes—
205#95C2414yes—
206#95C24112yes—
207C24—224no—
208C24—212no—
209C24—46no—
210C24—448no—
211C24—412no—
212C24—46no—
213C24—832no—
214C24C2×#41112yes—
215C24C2×#41112yes—
216C24C2×#4114yes—
217C24C2513yes—
218C24C2511yescharacteristic
219C24C2513yes—
220C2×Q8—412no—
221C22×C4—224no—
222C22×C4—248no—
223C22×C4—212no—
224C22×C4—212no—
225C22×C4—224no—
226C22×C4—224no—
227C22×C4—424no—
228C22×C4—412no—
229C22×C4—424no—
230C22×C4—412no—
231C22×C4—412no—
232C22×C4—848no—
233C22×C4C25112yes—
234C22×C4C25112yes—
235C22×C4C25112yes—
236C22×C4C2518yes—
237C22×C4C25112yes—
238C4⋊C4—412no—
239C4⋊C4—424no—
240C4⋊C4—412no—
241C4⋊C4—424no—
242C4⋊C4—424no—
243C4⋊C4—412no—
244C4⋊C4—424no—
245C22⋊C4—212no—
246C22⋊C4—212no—
247C22⋊C4—224no—
248C22⋊C4—224no—
249C22⋊C4—212no—
250C22⋊C4—212no—
251C22⋊C4—224no—
252C22⋊C4—248no—
253C22⋊C4—412no—
254C22⋊C4—412no—
255C22⋊C4—412no—
256C22⋊C4—46no—
257C22⋊C4—412no—
258C22⋊C4—46no—
259C42—424no—
260C42—424no—
261C23—212no—
262C23—26no—
263C23—212no—
264C23—26no—
265C23—248no—
266C23—224no—
267C23—412no—
268C23—424no—
269C23—412no—
270C23—448no—
271C23—824no—
272C23—864no—
273C23—88no—
274C23C22×#4112yes—
275C23C22×#4116yes—
276C23C22×#4116yes—
277C23#585112yes—
278C23C2611yescharacteristic
279Q8—164no—
280C2×C4—212no—
281C2×C4—212no—
282C2×C4—224no—
283C2×C4—224no—
284C2×C4—224no—
285C2×C4—224no—
286C2×C4—212no—
287C2×C4—212no—
288C2×C4—224no—
289C2×C4—448no—
290C2×C4—824no—
291C2×C4—824no—
292C2×C4—812no—
293C2×C4—812no—
294C2×C4—164no—
295C22—224no—
296C22—212no—
297C22—212no—
298C22—26no—
299C22—26no—
300C22—448no—
301C22—44no—
302C22—412no—
303C22—824no—
304C22—88no—
305C22#3491112yes—
306C22C23×#4111yescharacteristic
307C22C2×#58516yes—
308C4—412no—
309C4—412no—
310C4—48no—
311C4—412no—
312C4—412no—
313C2—212no—
314C2—46no—
315C2—42no—
316C2C2×#349113yes—
317C2#6302614yes—
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. #59196C2semidirect
2. #47196C2semidirect
3. #46838C2semidirect
4. #41426C2semidirect
6. #36013C2semidirect
12. #2835C22semidirect
13. #2834C22semidirect
14. #2745C22semidirect
15. #2744C22semidirect
16. #2739C22semidirect
17. #2738C22semidirect
18. #2652C22semidirect
20. #2575C22semidirect
21. #2573C22semidirect
22. #2569C22semidirect
23. #2527C22semidirect
24. #2523C22semidirect
25. #2517C22semidirect
26. #2515C22semidirect
27. #2505C22semidirect
28. #2503C22semidirect
29. #2493C22semidirect
30. #2493C22semidirect
31. #2484C22semidirect
38. #2438C22semidirect
40. #2401C22semidirect
42. #2368C22semidirect
43. #2344C22semidirect
44. #2343C22semidirect
70. #401C23semidirect
71. #401C23semidirect
72. #400C23semidirect
73. #400C23semidirect
74. #400C23semidirect
75. #400C23semidirect
76. #400C23semidirect
79. #398C23semidirect
80. #398C23semidirect
89. #395C23semidirect
92. #391C23semidirect
93. #389C23semidirect
98. #387C23semidirect
99. #387C23semidirect
100. #387C23semidirect
101. #387C23semidirect
102. #387C23semidirect
103. #387C23semidirect
104. #387C23semidirect
105. #387C23semidirect
116. #383C23semidirect
117. #382C23semidirect
135. C4×C4⋊C4C23semidirect
142. #376C23semidirect

Maximal subgroups

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

subgroupindex
#591962
#471962
#468382
#414262
#374752
#360132
C2×#24832
#341762