Finite Group Databasebeta

← #213840

Subgroups — #213840

Order 512.

1130 conjugacy classes of subgroups; only the 190 normal subgroups are shown.

normal subgroups only

This group has 1130 conjugacy classes of subgroups — too many to draw. Only its 190 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#20724C212yesmaximal, complemented
3#9502C214yesmaximal, complemented
4#9496C214yesmaximal, complemented
5#8746C212yesmaximal, complemented
6#8542C212yesmaximal
7C22×#121C2211yescharacteristic
8C2×#518C2211yescharacteristic
9#2484C412yescomplemented
10#2483C2212yes—
11#2482C412yescomplemented
12#2444—24no—
13#2443—24no—
14#2405C2212yes—
15#2312C2214yes—
16#2308C412yescomplemented
17#2308C2211yescharacteristic
18C2×#385C412yescomplemented
19#1819C2214yes—
20#1819C2214yes—
21#1799—24no—
22#1781C2214yes—
23#1781C2214yes—
24C4×#99—24no—
25#1304C2218yes—
26C23×D4C2×C411yescharacteristic
27C23×D4C2311yescharacteristic
28C23×D4C2311yescharacteristic
29C24×C4C2×C411yescharacteristic
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×C412yes—
41C22×C22⋊C4C2×C412yes—
42C22×C22⋊C4C2311yescharacteristic
43C22×C22⋊C4C2312yes—
44C2×#99—28no—
45C2×#99—44no—
46#400—22no—
47#399—22no—
48#398—22no—
49#397—22no—
50#397—24no—
51#394—24no—
52#388—24no—
53#388—48no—
54#387—24no—
55#386—24no—
56#386—22no—
57#386—22no—
58#386—22no—
59#386—24no—
60#385—22no—
61#385—24no—
62#385C2×C414yes—
63#385C2×C414yes—
64#381—22no—
65#381C2×C414yes—
66#381C2×C412yes—
67#380—24no—
68#379C2×C414yes—
69#379C2318yes—
70#379C2312yes—
71C4×C22⋊C4—24no—
72C2×#95—22no—
73#342—28no—
74#342—44no—
75C25C22×C411yescharacteristic
76C25C22×C411yescharacteristic
77C25C2411yescharacteristic
78C22×D4—24no—
79C22×D4—22no—
80C22×D4—22no—
81C22×D4—24no—
82C22×D4—42no—
83C22×D4—44no—
84C22×D4—44no—
85C22×D4—82no—
86C22×D4C2×D414yes—
87C23×C4—22no—
88C23×C4—22no—
89C23×C4—42no—
90C23×C4—82no—
91C23×C4C22⋊C412yes—
92C23×C4C22⋊C412yes—
93C23×C4C22×C411yescharacteristic
94C23×C4C22×C411yescharacteristic
95C23×C4C22×C411yescharacteristic
96C23×C4C22×C411yescharacteristic
97C23×C4C2×D412yes—
98C23×C4C2×D412yes—
99#123—84no—
100#121—416no—
101C4×D4—48no—
102C4×D4—48no—
103#117—44no—
104C2×C4⋊C4—24no—
105C2×C4⋊C4—22no—
106C2×C4⋊C4—24no—
107C2×C4⋊C4—42no—
108C2×C4⋊C4—44no—
109C2×C4⋊C4—84no—
110C2×C4⋊C4C22⋊C412yes—
111C2×C4⋊C4C22⋊C412yes—
112C2×C22⋊C4—28no—
113C2×C22⋊C4—22no—
114C2×C22⋊C4—22no—
115C2×C22⋊C4—24no—
116C2×C22⋊C4—22no—
117C2×C22⋊C4—28no—
118C2×C22⋊C4—22no—
119C2×C22⋊C4—22no—
120C2×C22⋊C4—24no—
121C2×C22⋊C4—24no—
122C2×C22⋊C4—28no—
123C2×C22⋊C4—28no—
124C2×C22⋊C4—44no—
125C2×C22⋊C4—42no—
126C2×C22⋊C4—44no—
127C2×C22⋊C4C22⋊C414yes—
128C2×C22⋊C4C22×C414yes—
129C2×C22⋊C4C22×C414yes—
130C2×C22⋊C4C2×D414yes—
131C2×C42—44no—
132C2×C42C22⋊C414yes—
133#99—416no—
134#99—88no—
135#95—24no—
136#95—24no—
137#95—24no—
138#95—24no—
139#95—22no—
140#95—22no—
141#95—44no—
142#95—44no—
143C24—22no—
144C24—24no—
145C24—21no—
146C24—22no—
147C24—42no—
148C24—42no—
149C24—81no—
150C24—81no—
151C24—81no—
152C24—81no—
153C24—81no—
154C24—81no—
155C24C2×C22⋊C412yes—
156C24C23×C411yescharacteristic
157C24C22×D412yes—
158C24H(5,𝔽2)14yes—
159C2×D4—28no—
160C2×D4—28no—
161C2×D4—44no—
162C2×D4—48no—
163C2×D4—44no—
164C2×D4—416no—
165C2×D4—44no—
166C2×D4—44no—
167C2×D4—44no—
168C2×D4—44no—
169C2×D4—82no—
170C2×D4—84no—
171C2×D4—88no—
172C22×C4—21no—
173C22×C4—22no—
174C22×C4—22no—
175C22×C4—24no—
176C22×C4—24no—
177C22×C4—21no—
178C22×C4—24no—
179C22×C4—21no—
180C22×C4—22no—
181C22×C4—22no—
182C22×C4—42no—
183C22×C4—42no—
184C22×C4—42no—
185C22×C4—42no—
186C22×C4—44no—
187C22×C4—44no—
188C22×C4—44no—
189C22×C4—42no—
190C22×C4—48no—
191C22×C4—84no—
192C22×C4—82no—
193C22×C4—82no—
194C22×C4—82no—
195C22×C4—82no—
196C22×C4—81no—
197C22×C4—82no—
198C22×C4—82no—
199C22×C4—82no—
200C22×C4—84no—
201C22×C4C2×C22⋊C412yes—
202C22×C4C2×C22⋊C414yes—
203C22×C4C2×C22⋊C414yes—
204C4⋊C4—42no—
205C4⋊C4—42no—
206C4⋊C4—44no—
207C4⋊C4—48no—
208C4⋊C4—84no—
209C4⋊C4—164no—
210C22⋊C4—28no—
211C22⋊C4—28no—
212C22⋊C4—44no—
213C22⋊C4—44no—
214C22⋊C4—44no—
215C22⋊C4—44no—
216C22⋊C4—44no—
217C22⋊C4—416no—
218C22⋊C4—84no—
219C22⋊C4—82no—
220C22⋊C4—84no—
221C22⋊C4—82no—
222C22⋊C4—88no—
223C22⋊C4—88no—
224C42—44no—
225C42—164no—
226C23—28no—
227C23—22no—
228C23—22no—
229C23—22no—
230C23—24no—
231C23—41no—
232C23—41no—
233C23—42no—
234C23—42no—
235C23—44no—
236C23—44no—
237C23—44no—
238C23—42no—
239C23—42no—
240C23—42no—
241C23—42no—
242C23—44no—
243C23—41no—
244C23—42no—
245C23—82no—
246C23—82no—
247C23—82no—
248C23—82no—
249C23—88no—
250C23—164no—
251C23—164no—
252C23#41014yes—
253C23C22×C22⋊C411yescharacteristic
254C23#51812yes—
255C23#53414yes—
256D4—416no—
257D4—88no—
258D4—88no—
259C2×C4—22no—
260C2×C4—24no—
261C2×C4—22no—
262C2×C4—24no—
263C2×C4—24no—
264C2×C4—24no—
265C2×C4—41no—
266C2×C4—42no—
267C2×C4—44no—
268C2×C4—41no—
269C2×C4—44no—
270C2×C4—44no—
271C2×C4—44no—
272C2×C4—44no—
273C2×C4—42no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—44no—
277C2×C4—42no—
278C2×C4—84no—
279C2×C4—84no—
280C2×C4—88no—
281C2×C4—88no—
282C2×C4—82no—
283C2×C4—82no—
284C2×C4—88no—
285C2×C4—88no—
286C2×C4—88no—
287C2×C4—162no—
288C2×C4—162no—
289C22—21no—
290C22—21no—
291C22—22no—
292C22—24no—
293C22—24no—
294C22—22no—
295C22—22no—
296C22—24no—
297C22—24no—
298C22—44no—
299C22—44no—
300C22—44no—
301C22—42no—
302C22—42no—
303C22—84no—
304C22—84no—
305C22—88no—
306C22—81no—
307C22—81no—
308C22—81no—
309C22—164no—
310C22—164no—
311C22#230811yescharacteristic
312C22C2×#41012yes—
313C4—42no—
314C4—44no—
315C4—44no—
316C4—42no—
317C4—84no—
318C4—81no—
319C4—81no—
320C4—168no—
321C2—22no—
322C2—44no—
323C2—42no—
324C2—44no—
325C2—161no—
326C2#2174711yescharacteristic
327C2#2177611yescharacteristic
328C2#2177711yescharacteristic
subgroupG/H
center, socleC22#2308
commutatorC24C23×C4
Frattini, cosocleC25C24
prime radicalC22×#121C22
LCSG ▷ C24 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C25 ◁ G
LECSG ▷ C25 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20724C2semidirect
3. #9502C2semidirect
4. #9496C2semidirect
5. #8746C2semidirect
9. #2484C4semidirect
11. #2482C4semidirect
16. #2308C4semidirect
18. C2×#385C4semidirect

Maximal subgroups

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

subgroupindex
#353512
#207242
#95022
#94962
#87462
#85422