Finite Group Databasebeta

← #391710

Subgroups — #391710

Order 512.

1956 conjugacy classes of subgroups; only the 234 normal subgroups are shown.

normal subgroups only

This group has 1956 conjugacy classes of subgroups — too many to draw. Only its 234 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
1#35446C211yesmaximal, characteristic
2#20585C212yesmaximal
3#11172C214yesmaximal, complemented
4#11169C212yesmaximal, complemented
5#10992C214yesmaximal, complemented
6#10966C212yesmaximal, complemented
7C22×H(5,𝔽2)C2211yescharacteristic
8C22×#123C2212yes—
9C22×#123C2211yescharacteristic
10C22×#99C2214yes—
11C22×#99C2214yes—
12#2406—24no—
13#2406C414yescomplemented
14#2405—24no—
15#2405C414yescomplemented
16#2308C2212yes—
17#2308C2214yes—
18#2308C2211yescharacteristic
19C2×#381C2214yescomplemented
20#1782—24no—
21#1781—24no—
22C2×#342C2218yescomplemented
23C2×#342C2214yescomplemented
24C2×H(5,𝔽2)—44no—
25C2×H(5,𝔽2)—48no—
26C22×#41C2×C412yes—
27C22×#41D414yes—
28C23×D4C2×C412yes—
29C23×D4D414yes—
30C23×D4C2311yescharacteristic
31C23×D4C2312yes—
32C24×C4C2311yescharacteristic
33C2×#123—24no—
34C2×#123—24no—
35C2×#123—48no—
36C2×#123—44no—
37C2×#123C2×C418yes—
38C22×C4⋊C4—22no—
39C22×C4⋊C4C2×C412yes—
40C22×C22⋊C4—22no—
41C22×C22⋊C4C2×C412yes—
42C22×C22⋊C4C2318yes—
43C22×C22⋊C4C2311yescharacteristic
44C22×C22⋊C4C2312yes—
45C2×#99—28no—
46C2×#99—28no—
47C2×#99—48no—
48C2×#99—48no—
49#395—24no—
50#394—24no—
51#394—28no—
52#386—24no—
53#386—24no—
54#386—48no—
55#386C2×C418yes—
56#385—28no—
57#385—24no—
58#381—22no—
59#381—24no—
60#381—48no—
61#381C2×C412yes—
62#379—24no—
63#379—22no—
64#379—24no—
65#379C2×C412yes—
66C2×#95—44no—
67#342—48no—
68#342—44no—
69C25C22⋊C412yes—
70C25C22×C411yescharacteristic
71C25C2×D412yes—
72C25C2411yescharacteristic
73H(5,𝔽2)—416no—
74H(5,𝔽2)—432no—
75H(5,𝔽2)—416no—
76C2×#41—44no—
77C2×#41—48no—
78C2×#41—48no—
79C2×#41—48no—
80C2×#41—48no—
81C22×Q8C22⋊C412yes—
82C22×D4—24no—
83C22×D4—28no—
84C22×D4—28no—
85C22×D4—28no—
86C22×D4—28no—
87C22×D4—48no—
88C22×D4—44no—
89C22×D4—48no—
90C22×D4—48no—
91C22×D4—48no—
92C22×D4C22⋊C414yes—
93C22×D4C22⋊C412yes—
94C22×D4C22×C414yes—
95C22×D4C2×D414yes—
96C22×D4C2×D414yes—
97C23×C4—22no—
98C23×C4—24no—
99C23×C4—28no—
100C23×C4—22no—
101C23×C4—28no—
102C23×C4C22⋊C412yes—
103C23×C4C22⋊C414yes—
104C23×C4C22×C412yes—
105C23×C4C22×C411yescharacteristic
106C23×C4C22×C412yes—
107C23×C4C2×D412yes—
108#123—416no—
109#123—416no—
110#123—416no—
111C2×C4⋊C4—28no—
112C2×C4⋊C4—24no—
113C2×C4⋊C4—44no—
114C2×C4⋊C4—48no—
115C2×C22⋊C4—28no—
116C2×C22⋊C4—28no—
117C2×C22⋊C4—24no—
118C2×C22⋊C4—22no—
119C2×C22⋊C4—22no—
120C2×C22⋊C4—24no—
121C2×C22⋊C4—28no—
122C2×C22⋊C4—28no—
123C2×C22⋊C4—28no—
124C2×C22⋊C4—28no—
125C2×C22⋊C4—28no—
126C2×C22⋊C4—28no—
127C2×C22⋊C4—28no—
128C2×C22⋊C4—24no—
129C2×C22⋊C4—44no—
130C2×C22⋊C4—48no—
131C2×C22⋊C4—48no—
132C2×C22⋊C4C22×C414yes—
133#99—416no—
134#99—416no—
135#95—28no—
136#95—24no—
137#95—48no—
138#95—416no—
139C24—22no—
140C24—24no—
141C24—24no—
142C24—24no—
143C24—24no—
144C24—44no—
145C24—42no—
146C24—82no—
147C24—84no—
148C24—82no—
149C24—84no—
150C24C2×C22⋊C414yes—
151C24C2×C22⋊C412yes—
152C24C23×C411yescharacteristic
153C24C22×D412yes—
154C24H(5,𝔽2)14yes—
155C24H(5,𝔽2)18yes—
156#41—432no—
157#41—464no—
158C2×Q8—44no—
159C2×Q8—42no—
160C2×D4—28no—
161C2×D4—28no—
162C2×D4—28no—
163C2×D4—28no—
164C2×D4—28no—
165C2×D4—28no—
166C2×D4—28no—
167C2×D4—28no—
168C2×D4—28no—
169C2×D4—44no—
170C2×D4—44no—
171C2×D4—44no—
172C2×D4—42no—
173C2×D4—44no—
174C2×D4—432no—
175C2×D4—464no—
176C2×D4—44no—
177C2×D4—48no—
178C2×D4—48no—
179C2×D4—416no—
180C2×D4—44no—
181C2×D4—48no—
182C22×C4—22no—
183C22×C4—24no—
184C22×C4—24no—
185C22×C4—24no—
186C22×C4—24no—
187C22×C4—22no—
188C22×C4—24no—
189C22×C4—22no—
190C22×C4—24no—
191C22×C4—28no—
192C22×C4—24no—
193C22×C4—44no—
194C22×C4—44no—
195C22×C4—44no—
196C22×C4—44no—
197C22×C4—42no—
198C22×C4—44no—
199C22×C4—44no—
200C22×C4—44no—
201C22×C4—44no—
202C22×C4—44no—
203C22×C4—48no—
204C22×C4—44no—
205C22×C4—48no—
206C22×C4—48no—
207C22×C4—48no—
208C22×C4—84no—
209C22×C4—82no—
210C22×C4—816no—
211C22×C4C2×C22⋊C412yes—
212C22×C4C2×C22⋊C414yes—
213C4⋊C4—416no—
214C4⋊C4—48no—
215C22⋊C4—28no—
216C22⋊C4—416no—
217C22⋊C4—48no—
218C22⋊C4—44no—
219C22⋊C4—44no—
220C22⋊C4—48no—
221C22⋊C4—48no—
222C22⋊C4—48no—
223C22⋊C4—48no—
224C22⋊C4—88no—
225C22⋊C4—84no—
226C22⋊C4—168no—
227C23—24no—
228C23—22no—
229C23—24no—
230C23—24no—
231C23—24no—
232C23—48no—
233C23—42no—
234C23—44no—
235C23—48no—
236C23—44no—
237C23—48no—
238C23—44no—
239C23—48no—
240C23—88no—
241C23—84no—
242C23—88no—
243C23—88no—
244C23—164no—
245C23—164no—
246C23—164no—
247C23—168no—
248C23C22×C22⋊C411yescharacteristic
249C23#51812yes—
250C23#51814yes—
251C23#53414yes—
252C23#53414yes—
253C23#53414yes—
254C23#56014yes—
255C23#56018yes—
256C23H(3,𝔽4)14yes—
257Q8—48no—
258D4—416no—
259D4—48no—
260D4—416no—
261D4—416no—
262D4—416no—
263C2×C4—24no—
264C2×C4—24no—
265C2×C4—24no—
266C2×C4—22no—
267C2×C4—24no—
268C2×C4—48no—
269C2×C4—416no—
270C2×C4—44no—
271C2×C4—48no—
272C2×C4—48no—
273C2×C4—44no—
274C2×C4—44no—
275C2×C4—44no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—48no—
279C2×C4—88no—
280C2×C4—88no—
281C2×C4—84no—
282C2×C4—816no—
283C2×C4—816no—
284C2×C4—88no—
285C2×C4—168no—
286C22—22no—
287C22—28no—
288C22—24no—
289C22—44no—
290C22—44no—
291C22—48no—
292C22—44no—
293C22—44no—
294C22—44no—
295C22—48no—
296C22—48no—
297C22—44no—
298C22—44no—
299C22—42no—
300C22—84no—
301C22—84no—
302C22—168no—
303C22—164no—
304C22—168no—
305C22#230812yes—
306C22#230811yescharacteristic
307C22#237514yes—
308C22#281218yes—
309C4—44no—
310C4—42no—
311C4—84no—
312C4—82no—
313C4—82no—
314C4—168no—
315C2—22no—
316C2—28no—
317C2—24no—
318C2—84no—
319C2—82no—
320C2#1465711yescharacteristic
321C2#2177712yes—
subgroupG/H
center, socleC22#2308
commutatorC24C23×C4
Frattini, cosocleC25C24
prime radicalC22×H(5,𝔽2)C22
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C25 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #11172C2semidirect
4. #11169C2semidirect
5. #10992C2semidirect
6. #10966C2semidirect
13. #2406C4semidirect
15. #2405C4semidirect
19. C2×#381C22semidirect
22. C2×#342C22semidirect
23. C2×#342C22semidirect

Maximal subgroups

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

subgroupindex
#354462
#205852
#111722
#111692
#109922
#109662