Finite Group Databasebeta

← #513749

Subgroups — #513749

Order 512.

8698 conjugacy classes of subgroups.

The subgroup lattice has 327 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#520C212yesmaximal
2C22×C4×#41C211yesmaximal, characteristic
3Q8×#95C214yesmaximal, complemented
4D4×#95C2112yesmaximal, complemented
5#13795C216yesmaximal, complemented
6#13756C213yesmaximal, complemented
7C2×C4⋊#95C213yesmaximal, complemented
8C23×#41C2211yescharacteristic
9C2×#520—216no—
10C2×#520—216no—
11C2×#520C4132yescomplemented
12C2×C4×#41—216no—
13C2×C4×#41C4116yescomplemented
14C22×C4×Q8C2212yes—
15C22×C4×Q8C2214yes—
16C22×C4×D4C2216yes—
17C22×C4×D4C22112yes—
18C22×#117C2213yes—
19C22×#117C2216yes—
20C23×C4⋊C4C2216yes—
21C23×C42C2213yes—
22C22×#95C22112yes—
23C4⋊#95C2213yes—
24C4⋊#95C2216yes—
25C4⋊#95C22112yes—
26C4⋊#95C2211yescharacteristic
27C4⋊#95C2216yes—
28C4⋊#95C22124yes—
29C22⋊#95C22112yes—
30C22⋊#95C22124yes—
31C4×#95C22112yes—
32C22×#41C2×C418yes—
33C22×#41C2×C4116yes—
34C22×#41D4116yes—
35C22×#41D418yes—
36C22×#41Q818yes—
37C23×Q8C2311yescharacteristic
38C23×D4C2313yes—
39C24×C4C2316yes—
40C24×C4C23112yes—
41C24×C4C2313yes—
42#520—2256no—
43C4×#41—2128no—
44C2×C4×Q8—216no—
45C2×C4×Q8—216no—
46C2×C4×Q8—216no—
47C2×C4×Q8C2×C4116yescomplemented
48C2×C4×Q8C2×C4132yescomplemented
49C2×C4×D4—248no—
50C2×C4×D4—248no—
51C2×C4×D4—248no—
52C2×C4×D4C2×C4148yescomplemented
53C2×C4×D4C2×C4196yescomplemented
54C2×#117—224no—
55C2×#117—224no—
56C2×#117—224no—
57C2×#117C2×C4124yescomplemented
58C2×#117C2×C4148yescomplemented
59C22×C4⋊C4—224no—
60C22×C4⋊C4—224no—
61C22×C4⋊C4C2×C4148yescomplemented
62C22×C4⋊C4C2316yes—
63C22×C4⋊C4C23112yes—
64C22×C4⋊C4C2312yes—
65C22×C4⋊C4C2316yes—
66C22×C22⋊C4C23112yes—
67C22×C22⋊C4C2316yes—
68C22×C42—224no—
69C22×C42C2×C4124yescomplemented
70C22×C42C2311yescharacteristic
71C22×C42C2313yes—
72C22×C42C2316yes—
73C22×C42C23112yes—
74C2×#95—424no—
75C2×#95C2314yes—
76C2×#95C23112yes—
77C2×#95C23124yes—
78C2×#95C23124yes—
79C25C2413yes—
80C2×#41C42164yes—
81C2×#41C22⋊C4164yes—
82C2×#41C22⋊C41128yes—
83C2×#41C4⋊C4164yes—
84C2×#41C4⋊C41128yes—
85C22×Q8C22×C414yes—
86C22×Q8C22×C418yes—
87C22×Q8C2×D418yes—
88C22×Q8C2×D414yes—
89C22×Q8C2×Q814yes—
90C22×D4C22×C4112yes—
91C22×D4C22×C4124yes—
92C22×D4C2×D4124yes—
93C22×D4C2×D4112yes—
94C22×D4C2×Q8112yes—
95C23×C4—224no—
96C23×C4—224no—
97C23×C4—224no—
98C23×C4—212no—
99C23×C4—212no—
100C23×C4—412no—
101C23×C4—424no—
102C23×C4—424no—
103C23×C4C22×C4124yes—
104C23×C4C22×C4148yes—
105C23×C4C22×C4124yes—
106C23×C4C22×C4112yes—
107C23×C4C2×D4112yes—
108C23×C4C2×D4124yes—
109C23×C4C2×Q8112yes—
110C23×C4C2414yes—
111C23×C4C2416yes—
112C23×C4C24112yes—
113C23×C4C2412yes—
114C23×C4C2411yescharacteristic
115C23×C4C2413yes—
116C4×Q8—264no—
117C4×Q8—2128no—
118C4×D4—2192no—
119C4×D4—2384no—
120#117—296no—
121#117—2192no—
122C2×C4⋊C4—2192no—
123C2×C4⋊C4—212no—
124C2×C4⋊C4—212no—
125C2×C4⋊C4—224no—
126C2×C4⋊C4—224no—
127C2×C4⋊C4—24no—
128C2×C4⋊C4—24no—
129C2×C4⋊C4—224no—
130C2×C4⋊C4C22×C4124yes—
131C2×C4⋊C4C22×C4148yes—
132C2×C4⋊C4C22×C418yes—
133C2×C4⋊C4C22×C4124yes—
134C2×C22⋊C4—224no—
135C2×C22⋊C4—224no—
136C2×C22⋊C4—224no—
137C2×C22⋊C4C22×C4148yes—
138C2×C22⋊C4C22×C4124yes—
139C2×C42—296no—
140C2×C42—212no—
141C2×C42—224no—
142C2×C42—224no—
143C2×C42—224no—
144C2×C42—24no—
145C2×C42C22×C4112yes—
146C2×C42C22×C4148yes—
147C2×C42C22×C4124yes—
148C2×C42C22×C414yes—
149#95—224no—
150#95—28no—
151#95—424no—
152#95—424no—
153C24—26no—
154C24—26no—
155C24—212no—
156C24C23×C416yes—
157C24C23×C4112yes—
158C24C22×D4112yes—
159C24C22×D416yes—
160C24C22×Q816yes—
161C24C2511yescharacteristic
162#41#951512yescomplemented
163C2×Q8C2×C42116yes—
164C2×Q8C2×C22⋊C4116yes—
165C2×Q8C2×C22⋊C4132yes—
166C2×Q8C2×C4⋊C4116yes—
167C2×Q8C2×C4⋊C4132yes—
168C2×D4C2×C42148yes—
169C2×D4C2×C22⋊C4148yes—
170C2×D4C2×C22⋊C4196yes—
171C2×D4C2×C4⋊C4148yes—
172C2×D4C2×C4⋊C4196yes—
173C22×C4—24no—
174C22×C4—24no—
175C22×C4—248no—
176C22×C4—24no—
177C22×C4—24no—
178C22×C4—212no—
179C22×C4—212no—
180C22×C4—212no—
181C22×C4—224no—
182C22×C4—26no—
183C22×C4—28no—
184C22×C4—26no—
185C22×C4—28no—
186C22×C4—212no—
187C22×C4—224no—
188C22×C4—296no—
189C22×C4—248no—
190C22×C4—424no—
191C22×C4—496no—
192C22×C4—448no—
193C22×C4—448no—
194C22×C4—848no—
195C22×C4—824no—
196C22×C4—824no—
197C22×C4C2×C42148yes—
198C22×C4C2×C22⋊C4148yes—
199C22×C4C2×C22⋊C4196yes—
200C22×C4C2×C4⋊C4196yes—
201C22×C4C2×C4⋊C4148yes—
202C22×C4C23×C418yes—
203C22×C4C23×C4112yes—
204C22×C4C23×C4124yes—
205C22×C4C23×C414yes—
206C22×C4C23×C412yes—
207C22×C4C23×C416yes—
208C22×C4C23×C4112yes—
209C22×C4C23×C414yes—
210C22×C4C22×D414yes—
211C22×C4C22×D4112yes—
212C22×C4C22×D416yes—
213C22×C4C22×D412yes—
214C22×C4C22×Q812yes—
215C22×C4C22×Q816yes—
216C22×C4H(5,𝔽2)14yes—
217C22×C42−1+414yes—
218C4⋊C4—248no—
219C4⋊C4—296no—
220C4⋊C4—216no—
221C4⋊C4—248no—
222C22⋊C4—296no—
223C22⋊C4—248no—
224C42—28no—
225C42—224no—
226C42—248no—
227C42—296no—
228C23—212no—
229C23—212no—
230C23—224no—
231C23—224no—
232C23—224no—
233C23—224no—
234C23—26no—
235C23—26no—
236C23—212no—
237C23—212no—
238C23—212no—
239C23C22×C42112yes—
240C23C22×C22⋊C4112yes—
241C23C22×C22⋊C4124yes—
242C23C22×C4⋊C4112yes—
243C23C22×C4⋊C4124yes—
244C23C24×C411yescharacteristic
245C23C24×C412yes—
246C23C23×D412yes—
247C23C23×D411yescharacteristic
248C23C23×Q811yescharacteristic
249C23C2×H(5,𝔽2)12yes—
250C23C2×2−1+412yes—
251C23#58514yes—
252Q8C2×#95164yescomplemented
253D4C2×#951192yescomplemented
254C2×C4—24no—
255C2×C4—24no—
256C2×C4—28no—
257C2×C4—28no—
258C2×C4—28no—
259C2×C4—212no—
260C2×C4—224no—
261C2×C4—212no—
262C2×C4—212no—
263C2×C4—248no—
264C2×C4—216no—
265C2×C4—496no—
266C2×C4—412no—
267C2×C4—424no—
268C2×C4—48no—
269C2×C4—412no—
270C2×C4—412no—
271C2×C4—44no—
272C2×C4—44no—
273C2×C4—412no—
274C2×C4—896no—
275C2×C4C2×#951192yescomplemented
276C2×C4C22×C4214yes—
277C2×C4C22×C42112yes—
278C2×C4C22×C22⋊C414yes—
279C2×C4C22×C22⋊C4112yes—
280C2×C4C22×C22⋊C4124yes—
281C2×C4C22×C22⋊C418yes—
282C2×C4C22×C4⋊C414yes—
283C2×C4C22×C4⋊C418yes—
284C2×C4C22×C4⋊C4112yes—
285C2×C4C22×C4⋊C4124yes—
286C2×C4#52018yescomplemented
287C22—224no—
288C22—212no—
289C22—212no—
290C22—248no—
291C22—224no—
292C22—224no—
293C22—224no—
294C22C22×#95124yes—
295C22C23×C4211yescharacteristic
296C22C23×C22⋊C411yescharacteristic
297C22C23×C22⋊C412yes—
298C22C23×C4⋊C411yescharacteristic
299C22C23×C4⋊C412yes—
300C22C2×#52012yes—
301C22#342812yes—
302C22C4×H(5,𝔽2)14yes—
303C22C4×2−1+414yes—
304C22#348814yes—
305C22#348914yes—
306C22#349811yescharacteristic
307C22#349911yescharacteristic
308C22#350012yes—
309C22#350312yes—
310C22#350412yes—
311C4—28no—
312C4—412no—
313C4—424no—
314C4—48no—
315C4C22×#9518yes—
316C4C22×#95124yes—
317C2—224no—
318C2C23×#9511yescharacteristic
319C2C4×#52012yes—
320C2#3417512yes—
321C2#3418311yescharacteristic
322C2#3418411yescharacteristic
323C2#3425014yes—
324C2#3425812yes—
325C2#3425912yes—
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC22C23×C42
prime radicalC23×#41C22
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
3. Q8×#95C2semidirect
4. D4×#95C2semidirect
5. #13795C2semidirect
6. #13756C2semidirect
7. C2×C4⋊#95C2semidirect
11. C2×#520C4semidirect
13. C2×C4×#41C4semidirect
47. C2×C4×Q8C2×C4semidirect
48. C2×C4×Q8C2×C4semidirect
52. C2×C4×D4C2×C4semidirect
53. C2×C4×D4C2×C4semidirect
57. C2×#117C2×C4semidirect
58. C2×#117C2×C4semidirect
61. C22×C4⋊C4C2×C4semidirect
69. C22×C42C2×C4semidirect
162. #41#95semidirect
252. Q8C2×#95semidirect
253. D4C2×#95semidirect
275. C2×C4C2×#95semidirect
286. C2×C4#520semidirect

Maximal subgroups

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

subgroupindex
C22×#5202
C22×C4×#412
Q8×#952
D4×#952
#137952
#137562
C2×C4⋊#952