Finite Group Databasebeta

← #375670

Subgroups — #375670

Order 512.

1418 conjugacy classes of subgroups; only the 250 normal subgroups are shown.

normal subgroups only

This group has 1418 conjugacy classes of subgroups — too many to draw. Only its 250 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#20392C212yesmaximal, complemented
3#11169C212yesmaximal, complemented
4#10980C212yesmaximal
5#10938C214yesmaximal, complemented
6#10918C214yesmaximal, complemented
7C22×#121C2211yescharacteristic
8C2×#518C2211yescharacteristic
9C22×#99C2214yes—
10C22×#99C2214yes—
11#2484C412yescomplemented
12#2483C412yescomplemented
13#2482C412yescomplemented
14#2444—24no—
15#2443—24no—
16#2405C412yescomplemented
17#2308C2211yescharacteristic
18#2308C2212yes—
19C2×#385C2212yes—
20#2273C2214yes—
21C4×#99—24no—
22C4×#99—24no—
23C2×#342C2218yes—
24C2×#342C2214yes—
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⋊C4C2312yes—
43C22×C22⋊C4C2318yes—
44C22×C22⋊C4C2312yes—
45C2×#99—28no—
46C2×#99—216no—
47C2×#99—216no—
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#385C2×C414yes—
65#385C2×C414yes—
66#381—22no—
67#381C2×C412yes—
68#381C2×C414yes—
69#379C2×C414yes—
70#379C2×C412yes—
71C4×C22⋊C4—24no—
72C4×C22⋊C4—24no—
73C4×C22⋊C4—48no—
74C2×#95—22no—
75C2×#95—44no—
76#342—28no—
77#342—216no—
78C25C22×C411yescharacteristic
79C25C22×C411yescharacteristic
80C25C2411yescharacteristic
81C22×D4—28no—
82C22×D4—24no—
83C22×D4—22no—
84C22×D4—28no—
85C22×D4—24no—
86C22×D4—22no—
87C22×D4—42no—
88C22×D4—82no—
89C22×D4C2×D414yes—
90C23×C4—22no—
91C23×C4—24no—
92C23×C4—22no—
93C23×C4—28no—
94C23×C4—82no—
95C23×C4C22⋊C412yes—
96C23×C4C22⋊C412yes—
97C23×C4C22×C411yescharacteristic
98C23×C4C22×C411yescharacteristic
99C23×C4C22×C411yescharacteristic
100C23×C4C22×C411yescharacteristic
101C23×C4C2×D412yes—
102C23×C4C2×D412yes—
103#123—84no—
104#121—416no—
105C4×D4—48no—
106C4×D4—48no—
107#117—44no—
108C2×C4⋊C4—24no—
109C2×C4⋊C4—22no—
110C2×C4⋊C4—24no—
111C2×C4⋊C4—42no—
112C2×C4⋊C4—84no—
113C2×C4⋊C4C22⋊C412yes—
114C2×C4⋊C4C22⋊C412yes—
115C2×C22⋊C4—28no—
116C2×C22⋊C4—28no—
117C2×C22⋊C4—28no—
118C2×C22⋊C4—28no—
119C2×C22⋊C4—28no—
120C2×C22⋊C4—28no—
121C2×C22⋊C4—22no—
122C2×C22⋊C4—22no—
123C2×C22⋊C4—28no—
124C2×C22⋊C4—22no—
125C2×C22⋊C4—22no—
126C2×C22⋊C4—22no—
127C2×C22⋊C4—24no—
128C2×C22⋊C4—24no—
129C2×C22⋊C4—44no—
130C2×C22⋊C4—42no—
131C2×C22⋊C4—42no—
132C2×C22⋊C4C22⋊C414yes—
133C2×C22⋊C4C22×C414yes—
134C2×C22⋊C4C22×C414yes—
135C2×C22⋊C4C2×D414yes—
136C2×C22⋊C4#41116yes—
137C2×C42—44no—
138C2×C42—44no—
139C2×C42C22⋊C414yes—
140#99—232no—
141#99—416no—
142#95—24no—
143#95—24no—
144#95—24no—
145#95—24no—
146#95—22no—
147#95—22no—
148#95—44no—
149#95—44no—
150#95—88no—
151C24—22no—
152C24—24no—
153C24—21no—
154C24—22no—
155C24—24no—
156C24—81no—
157C24—81no—
158C24—81no—
159C24—81no—
160C24—81no—
161C24—81no—
162C24C2×C22⋊C412yes—
163C24C23×C411yescharacteristic
164C24C22×D412yes—
165C24C2×#4118yes—
166C24H(5,𝔽2)14yes—
167C2×D4—28no—
168C2×D4—28no—
169C2×D4—28no—
170C2×D4—44no—
171C2×D4—44no—
172C2×D4—48no—
173C2×D4—44no—
174C2×D4—44no—
175C2×D4—44no—
176C2×D4—44no—
177C2×D4—44no—
178C2×D4—44no—
179C2×D4—48no—
180C2×D4—44no—
181C2×D4—84no—
182C2×D4—82no—
183C2×D4—84no—
184C22×C4—21no—
185C22×C4—24no—
186C22×C4—22no—
187C22×C4—22no—
188C22×C4—24no—
189C22×C4—24no—
190C22×C4—21no—
191C22×C4—24no—
192C22×C4—21no—
193C22×C4—24no—
194C22×C4—22no—
195C22×C4—22no—
196C22×C4—216no—
197C22×C4—42no—
198C22×C4—42no—
199C22×C4—42no—
200C22×C4—48no—
201C22×C4—44no—
202C22×C4—44no—
203C22×C4—48no—
204C22×C4—44no—
205C22×C4—48no—
206C22×C4—81no—
207C22×C4—82no—
208C22×C4—82no—
209C22×C4—82no—
210C22×C4—84no—
211C22×C4—82no—
212C22×C4—82no—
213C22×C4—82no—
214C22×C4—82no—
215C22×C4—88no—
216C22×C4—88no—
217C22×C4C2×C22⋊C412yes—
218C22×C4C2×C22⋊C414yes—
219C22×C4C2×C22⋊C414yes—
220C4⋊C4—42no—
221C4⋊C4—42no—
222C4⋊C4—44no—
223C4⋊C4—48no—
224C4⋊C4—84no—
225C22⋊C4—28no—
226C22⋊C4—28no—
227C22⋊C4—28no—
228C22⋊C4—216no—
229C22⋊C4—216no—
230C22⋊C4—44no—
231C22⋊C4—44no—
232C22⋊C4—44no—
233C22⋊C4—44no—
234C22⋊C4—48no—
235C22⋊C4—48no—
236C22⋊C4—84no—
237C22⋊C4—82no—
238C22⋊C4—84no—
239C22⋊C4—82no—
240C42—44no—
241C42—164no—
242C42—164no—
243C23—22no—
244C23—24no—
245C23—42no—
246C23—44no—
247C23—41no—
248C23—42no—
249C23—44no—
250C23—41no—
251C23—42no—
252C23—42no—
253C23—42no—
254C23—42no—
255C23—44no—
256C23—41no—
257C23—42no—
258C23—44no—
259C23—82no—
260C23—82no—
261C23—82no—
262C23—82no—
263C23—88no—
264C23—164no—
265C23—164no—
266C23C22×C22⋊C411yescharacteristic
267C23C4×#4114yes—
268C23#51812yes—
269C23#52514yes—
270C23#53214yes—
271C23#53414yes—
272C23#55114yes—
273C23#55318yes—
274C23#55914yes—
275D4—416no—
276D4—88no—
277D4—88no—
278C2×C4—22no—
279C2×C4—24no—
280C2×C4—22no—
281C2×C4—24no—
282C2×C4—24no—
283C2×C4—24no—
284C2×C4—41no—
285C2×C4—42no—
286C2×C4—41no—
287C2×C4—44no—
288C2×C4—44no—
289C2×C4—44no—
290C2×C4—44no—
291C2×C4—44no—
292C2×C4—42no—
293C2×C4—44no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—416no—
297C2×C4—44no—
298C2×C4—44no—
299C2×C4—42no—
300C2×C4—84no—
301C2×C4—84no—
302C2×C4—88no—
303C2×C4—88no—
304C2×C4—88no—
305C2×C4—88no—
306C2×C4—816no—
307C2×C4—162no—
308C2×C4—162no—
309C2×C4—168no—
310C22—21no—
311C22—22no—
312C22—24no—
313C22—21no—
314C22—24no—
315C22—22no—
316C22—44no—
317C22—48no—
318C22—44no—
319C22—44no—
320C22—44no—
321C22—44no—
322C22—42no—
323C22—42no—
324C22—42no—
325C22—84no—
326C22—84no—
327C22—81no—
328C22—81no—
329C22—81no—
330C22—164no—
331C22—164no—
332C22#229712yes—
333C22#230811yescharacteristic
334C22#237414yes—
335C22#260218yes—
336C4—42no—
337C4—44no—
338C4—44no—
339C4—42no—
340C4—84no—
341C4—816no—
342C4—81no—
343C4—81no—
344C2—22no—
345C2—28no—
346C2—24no—
347C2—82no—
348C2—161no—
349C2#1416711yescharacteristic
350C2#2177611yescharacteristic
351C2#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. #20392C2semidirect
3. #11169C2semidirect
5. #10938C2semidirect
6. #10918C2semidirect
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
#203922
#111692
#109802
#109382
#109182