Finite Group Databasebeta

← #391739

Subgroups — #391739

Order 512.

1330 conjugacy classes of subgroups; only the 234 normal subgroups are shown.

normal subgroups only

This group has 1330 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#35351C211yesmaximal, characteristic
2#20633C212yesmaximal, complemented
3#11222C214yesmaximal, complemented
4#11169C212yesmaximal, complemented
5#11046C214yesmaximal, complemented
6#10980C212yesmaximal
7C22×#121C2211yescharacteristic
8C2×#518C2211yescharacteristic
9C22×#99C2214yes—
10C22×#99C2214yes—
11#2484C412yescomplemented
12#2483C412yescomplemented
13#2482C412yescomplemented
14#2444—24no—
15#2443—24no—
16#2405C412yescomplemented
17#2315C2214yes—
18#2308C2212yes—
19#2308C2211yescharacteristic
20C2×#385C2212yes—
21#1799—24no—
22#1799—24no—
23C2×#342C2214yes—
24C2×#342C2218yes—
25C22⋊#95C2214yes—
26C23×D4C2×C411yescharacteristic
27C23×D4C2311yescharacteristic
28C23×D4C2311yescharacteristic
29C24×C4C2311yescharacteristic
30C2×#123—22no—
31C2×#121—24no—
32C2×#121—44no—
33#518—44no—
34C2×C4×D4D414yes—
35C2×C4×D4D414yes—
36C2×#117C2×C412yes—
37C22×C4⋊C4—22no—
38C22×C4⋊C4C2×C411yescharacteristic
39C22×C22⋊C4C2×C411yescharacteristic
40C22×C22⋊C4C2×C411yescharacteristic
41C22×C22⋊C4C2×C412yes—
42C22×C22⋊C4C2318yes—
43C22×C22⋊C4C2312yes—
44C22×C22⋊C4C2312yes—
45C2×#99—28no—
46C2×#99—48no—
47C2×#99—48no—
48C2×#99—44no—
49#400—22no—
50#399—22no—
51#398—22no—
52#397—22no—
53#397—24no—
54#394—24no—
55#388—24no—
56#387—24no—
57#386—24no—
58#386—22no—
59#386—22no—
60#386—22no—
61#386—24no—
62#385—22no—
63#385—24no—
64#385—48no—
65#385C2×C414yes—
66#385C2×C414yes—
67#381—22no—
68#381C2×C412yes—
69#381C2×C414yes—
70#380—24no—
71#380—24no—
72#379C2×C414yes—
73#379C2×C412yes—
74C2×#95—22no—
75C2×#95—44no—
76#342—28no—
77#342—48no—
78C25C22×C411yescharacteristic
79C25C22×C411yescharacteristic
80C25C2411yescharacteristic
81C22×D4—24no—
82C22×D4—24no—
83C22×D4—24no—
84C22×D4—22no—
85C22×D4—24no—
86C22×D4—24no—
87C22×D4—24no—
88C22×D4—22no—
89C22×D4—42no—
90C22×D4—82no—
91C22×D4C2×D414yes—
92C23×C4—22no—
93C23×C4—24no—
94C23×C4—22no—
95C23×C4—28no—
96C23×C4—82no—
97C23×C4C22⋊C412yes—
98C23×C4C22⋊C412yes—
99C23×C4C22×C411yescharacteristic
100C23×C4C22×C411yescharacteristic
101C23×C4C22×C411yescharacteristic
102C23×C4C22×C411yescharacteristic
103C23×C4C2×D412yes—
104C23×C4C2×D412yes—
105#123—84no—
106#121—416no—
107C4×D4—48no—
108C4×D4—48no—
109#117—44no—
110C2×C4⋊C4—24no—
111C2×C4⋊C4—22no—
112C2×C4⋊C4—24no—
113C2×C4⋊C4—42no—
114C2×C4⋊C4—44no—
115C2×C4⋊C4—44no—
116C2×C4⋊C4—84no—
117C2×C4⋊C4C22⋊C412yes—
118C2×C4⋊C4C22⋊C412yes—
119C2×C22⋊C4—28no—
120C2×C22⋊C4—24no—
121C2×C22⋊C4—24no—
122C2×C22⋊C4—22no—
123C2×C22⋊C4—22no—
124C2×C22⋊C4—22no—
125C2×C22⋊C4—28no—
126C2×C22⋊C4—28no—
127C2×C22⋊C4—28no—
128C2×C22⋊C4—28no—
129C2×C22⋊C4—28no—
130C2×C22⋊C4—28no—
131C2×C22⋊C4—22no—
132C2×C22⋊C4—22no—
133C2×C22⋊C4—24no—
134C2×C22⋊C4—24no—
135C2×C22⋊C4—44no—
136C2×C22⋊C4—42no—
137C2×C22⋊C4—42no—
138C2×C22⋊C4C22⋊C414yes—
139C2×C22⋊C4C22×C414yes—
140C2×C22⋊C4C22×C414yes—
141C2×C22⋊C4C2×D414yes—
142C2×C42C22⋊C414yes—
143#99—416no—
144#99—416no—
145#95—24no—
146#95—24no—
147#95—24no—
148#95—24no—
149#95—22no—
150#95—22no—
151#95—44no—
152#95—44no—
153#95—88no—
154C24—22no—
155C24—24no—
156C24—21no—
157C24—22no—
158C24—24no—
159C24—81no—
160C24—81no—
161C24—81no—
162C24—81no—
163C24—81no—
164C24—81no—
165C24C2×C22⋊C412yes—
166C24C23×C411yescharacteristic
167C24C22×D412yes—
168C24H(5,𝔽2)14yes—
169C24H(5,𝔽2)14yes—
170C242−1+414yes—
171C2×D4—28no—
172C2×D4—24no—
173C2×D4—24no—
174C2×D4—28no—
175C2×D4—44no—
176C2×D4—44no—
177C2×D4—48no—
178C2×D4—44no—
179C2×D4—44no—
180C2×D4—44no—
181C2×D4—42no—
182C2×D4—42no—
183C2×D4—44no—
184C2×D4—44no—
185C2×D4—48no—
186C2×D4—44no—
187C2×D4—84no—
188C2×D4—82no—
189C2×D4—84no—
190C22×C4—21no—
191C22×C4—22no—
192C22×C4—22no—
193C22×C4—24no—
194C22×C4—24no—
195C22×C4—24no—
196C22×C4—21no—
197C22×C4—24no—
198C22×C4—21no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—24no—
202C22×C4—42no—
203C22×C4—42no—
204C22×C4—42no—
205C22×C4—48no—
206C22×C4—44no—
207C22×C4—44no—
208C22×C4—44no—
209C22×C4—48no—
210C22×C4—48no—
211C22×C4—48no—
212C22×C4—81no—
213C22×C4—82no—
214C22×C4—82no—
215C22×C4—82no—
216C22×C4—84no—
217C22×C4—82no—
218C22×C4—82no—
219C22×C4—82no—
220C22×C4—82no—
221C22×C4—88no—
222C22×C4—88no—
223C22×C4C2×C22⋊C412yes—
224C22×C4C2×C22⋊C414yes—
225C22×C4C2×C22⋊C414yes—
226C4⋊C4—42no—
227C4⋊C4—42no—
228C4⋊C4—44no—
229C4⋊C4—48no—
230C4⋊C4—84no—
231C4⋊C4—164no—
232C4⋊C4—164no—
233C22⋊C4—28no—
234C22⋊C4—28no—
235C22⋊C4—28no—
236C22⋊C4—44no—
237C22⋊C4—44no—
238C22⋊C4—44no—
239C22⋊C4—42no—
240C22⋊C4—42no—
241C22⋊C4—48no—
242C22⋊C4—48no—
243C22⋊C4—48no—
244C22⋊C4—48no—
245C22⋊C4—84no—
246C22⋊C4—82no—
247C22⋊C4—84no—
248C22⋊C4—82no—
249C42—44no—
250C23—22no—
251C23—24no—
252C23—42no—
253C23—44no—
254C23—41no—
255C23—42no—
256C23—44no—
257C23—41no—
258C23—42no—
259C23—42no—
260C23—42no—
261C23—42no—
262C23—44no—
263C23—41no—
264C23—42no—
265C23—44no—
266C23—82no—
267C23—82no—
268C23—82no—
269C23—82no—
270C23—84no—
271C23—84no—
272C23—164no—
273C23—164no—
274C23C22×C22⋊C411yescharacteristic
275C23#51812yes—
276C23#52014yes—
277C23#53414yes—
278C23#53512yes—
279C23#53612yes—
280C23#53714yes—
281C23#56014yes—
282C23#56218yes—
283C23#56314yes—
284D4—48no—
285D4—48no—
286D4—88no—
287D4—84no—
288D4—84no—
289C2×C4—22no—
290C2×C4—24no—
291C2×C4—22no—
292C2×C4—24no—
293C2×C4—24no—
294C2×C4—24no—
295C2×C4—41no—
296C2×C4—42no—
297C2×C4—41no—
298C2×C4—44no—
299C2×C4—44no—
300C2×C4—44no—
301C2×C4—44no—
302C2×C4—44no—
303C2×C4—42no—
304C2×C4—44no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—44no—
308C2×C4—42no—
309C2×C4—44no—
310C2×C4—84no—
311C2×C4—84no—
312C2×C4—88no—
313C2×C4—88no—
314C2×C4—88no—
315C2×C4—88no—
316C2×C4—816no—
317C2×C4—88no—
318C2×C4—162no—
319C2×C4—162no—
320C2×C4—168no—
321C22—21no—
322C22—22no—
323C22—24no—
324C22—21no—
325C22—24no—
326C22—22no—
327C22—44no—
328C22—48no—
329C22—44no—
330C22—44no—
331C22—42no—
332C22—42no—
333C22—44no—
334C22—42no—
335C22—42no—
336C22—42no—
337C22—84no—
338C22—84no—
339C22—81no—
340C22—81no—
341C22—81no—
342C22—164no—
343C22—164no—
344C22#230811yescharacteristic
345C22#231112yes—
346C22#237614yes—
347C22#281314yes—
348C22#281414yes—
349C4—42no—
350C4—44no—
351C4—44no—
352C4—42no—
353C4—84no—
354C4—81no—
355C4—81no—
356C4—168no—
357C2—22no—
358C2—24no—
359C2—24no—
360C2—24no—
361C2—82no—
362C2—161no—
363C2#1465811yescharacteristic
364C2#2177611yescharacteristic
365C2#2177711yescharacteristic
subgroupG/H
center, socleC22#2308
commutatorC24C23×C4
Frattini, cosocleC25C24
prime radicalC22×#121C22
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
2. #20633C2semidirect
3. #11222C2semidirect
4. #11169C2semidirect
5. #11046C2semidirect
11. #2484C4semidirect
12. #2483C4semidirect
13. #2482C4semidirect
16. #2405C4semidirect

Maximal subgroups

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

subgroupindex
#353512
#206332
#112222
#111692
#110462
#109802