Finite Group Databasebeta

← #391735

Subgroups — #391735

Order 512.

1686 conjugacy classes of subgroups; only the 234 normal subgroups are shown.

normal subgroups only

This group has 1686 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#35350C211yesmaximal, characteristic
2#20633C212yesmaximal, complemented
3#11205C214yesmaximal, complemented
4#11166C212yesmaximal, complemented
5#11069C214yesmaximal, complemented
6#10965C212yesmaximal
7C22×#120C2211yescharacteristic
8C2×#518C2211yescharacteristic
9C22×#99C2214yes—
10#2482C414yescomplemented
11#2418—28no—
12#2405C414yescomplemented
13#2315C2212yes—
14#2315C2214yes—
15#2308C2211yescharacteristic
16C2×#379C2212yes—
17C2×#379C2214yes—
18#1799—28no—
19C2×#342C2214yes—
20C2×#342C2218yes—
21C2×#342C2214yes—
22C26C2311yescharacteristic
23C23×D4C2×C412yes—
24C23×D4C2311yescharacteristic
25C2×#123—24no—
26C2×#120—24no—
27C2×#120—44no—
28#518—44no—
29C2×C4×D4D418yes—
30C2×#117C2×C412yes—
31C22×C22⋊C4—24no—
32C22×C22⋊C4—44no—
33C22×C22⋊C4C2×C412yes—
34C22×C22⋊C4C2×C412yes—
35C22×C22⋊C4C2318yes—
36C22×C22⋊C4C2311yescharacteristic
37C22×C22⋊C4C2312yes—
38C22×C22⋊C4C2312yes—
39C2×#99—28no—
40C2×#99—48no—
41#399—24no—
42#397—24no—
43#394—28no—
44#388—28no—
45#386—24no—
46#386—24no—
47#385—24no—
48#385—48no—
49#385C2×C418yes—
50#381C2×C412yes—
51#380—24no—
52#380—28no—
53#379—24no—
54#379—48no—
55#379C2×C412yes—
56#379C2×C418yes—
57#342—28no—
58#342—48no—
59#342—44no—
60C25—22no—
61C25—24no—
62C25—24no—
63C25—24no—
64C25—84no—
65C25C22×C412yes—
66C25C22×C411yescharacteristic
67C25C2411yescharacteristic
68C22×D4—24no—
69C22×D4—24no—
70C22×D4—24no—
71C22×D4—24no—
72C22×D4—44no—
73C22×D4—84no—
74C22×D4C2×D414yes—
75C23×C4—22no—
76C23×C4—28no—
77C23×C4C22⋊C414yescomplemented
78C23×C4C22×C411yescharacteristic
79C23×C4C22×C412yes—
80C23×C4C2×D414yes—
81#123—88no—
82#120—416no—
83C4×D4—416no—
84#117—44no—
85C2×C4⋊C4—28no—
86C2×C4⋊C4—48no—
87C2×C4⋊C4C22⋊C414yescomplemented
88C2×C22⋊C4—28no—
89C2×C22⋊C4—24no—
90C2×C22⋊C4—24no—
91C2×C22⋊C4—22no—
92C2×C22⋊C4—22no—
93C2×C22⋊C4—24no—
94C2×C22⋊C4—28no—
95C2×C22⋊C4—24no—
96C2×C22⋊C4—28no—
97C2×C22⋊C4—28no—
98C2×C22⋊C4—28no—
99C2×C22⋊C4—28no—
100C2×C22⋊C4—28no—
101C2×C22⋊C4—28no—
102C2×C22⋊C4—24no—
103C2×C22⋊C4—28no—
104C2×C22⋊C4—44no—
105C2×C22⋊C4—44no—
106C2×C22⋊C4—416no—
107C2×C22⋊C4—88no—
108C2×C22⋊C4—816no—
109C2×C22⋊C4C22⋊C414yescomplemented
110C2×C22⋊C4C22×C414yes—
111C2×C22⋊C4C22×C414yes—
112C2×C22⋊C4C2×D414yes—
113C2×C42C22⋊C414yescomplemented
114#99—416no—
115#95—28no—
116#95—24no—
117#95—48no—
118C24—22no—
119C24—24no—
120C24—22no—
121C24—24no—
122C24—28no—
123C24—44no—
124C24—48no—
125C24—416no—
126C24—44no—
127C24—44no—
128C24—44no—
129C24—44no—
130C24—42no—
131C24—42no—
132C24—48no—
133C24—82no—
134C24—82no—
135C24—84no—
136C24—88no—
137C24—816no—
138C24—816no—
139C24—82no—
140C24C2×C22⋊C412yes—
141C24C23×C411yescharacteristic
142C24C22×D412yes—
143C24H(5,𝔽2)14yes—
144C24H(5,𝔽2)14yes—
145C242−1+414yes—
146C2×D4—28no—
147C2×D4—24no—
148C2×D4—24no—
149C2×D4—28no—
150C2×D4—44no—
151C2×D4—48no—
152C2×D4—44no—
153C2×D4—48no—
154C2×D4—48no—
155C2×D4—84no—
156C2×D4—88no—
157C2×D4—88no—
158C22×C4—22no—
159C22×C4—24no—
160C22×C4—28no—
161C22×C4—24no—
162C22×C4—22no—
163C22×C4—24no—
164C22×C4—44no—
165C22×C4—42no—
166C22×C4—42no—
167C22×C4—48no—
168C22×C4—48no—
169C22×C4—48no—
170C22×C4—84no—
171C22×C4—82no—
172C22×C4—84no—
173C22×C4—84no—
174C22×C4—816no—
175C22×C4C2×C22⋊C412yes—
176C22×C4C2×C22⋊C418yes—
177C4⋊C4—44no—
178C4⋊C4—88no—
179C4⋊C4—168no—
180C22⋊C4—28no—
181C22⋊C4—28no—
182C22⋊C4—28no—
183C22⋊C4—44no—
184C22⋊C4—48no—
185C22⋊C4—416no—
186C22⋊C4—44no—
187C22⋊C4—44no—
188C22⋊C4—48no—
189C22⋊C4—48no—
190C22⋊C4—48no—
191C22⋊C4—48no—
192C22⋊C4—88no—
193C22⋊C4—84no—
194C22⋊C4—832no—
195C42—44no—
196C23—24no—
197C23—28no—
198C23—22no—
199C23—24no—
200C23—44no—
201C23—48no—
202C23—42no—
203C23—44no—
204C23—42no—
205C23—44no—
206C23—48no—
207C23—44no—
208C23—416no—
209C23—416no—
210C23—416no—
211C23—42no—
212C23—44no—
213C23—48no—
214C23—44no—
215C23—84no—
216C23—816no—
217C23—84no—
218C23—84no—
219C23—816no—
220C23—88no—
221C23—88no—
222C23—832no—
223C23—88no—
224C23—88no—
225C23—88no—
226C23—84no—
227C23—84no—
228C23—164no—
229C23C22×C22⋊C411yescharacteristic
230C23#51812yes—
231C23#52014yes—
232C23#53414yes—
233C23#53512yes—
234C23#53612yes—
235C23#53714yes—
236C23#56014yes—
237C23#56218yes—
238C23#56314yes—
239D4—48no—
240D4—48no—
241D4—816no—
242C2×C4—22no—
243C2×C4—28no—
244C2×C4—24no—
245C2×C4—44no—
246C2×C4—48no—
247C2×C4—48no—
248C2×C4—44no—
249C2×C4—42no—
250C2×C4—44no—
251C2×C4—88no—
252C2×C4—84no—
253C2×C4—816no—
254C2×C4—816no—
255C2×C4—88no—
256C2×C4—164no—
257C2×C4—162no—
258C2×C4—162no—
259C2×C4—168no—
260C22—22no—
261C22—28no—
262C22—24no—
263C22—44no—
264C22—42no—
265C22—42no—
266C22—48no—
267C22—44no—
268C22—44no—
269C22—44no—
270C22—48no—
271C22—416no—
272C22—44no—
273C22—44no—
274C22—42no—
275C22—84no—
276C22—84no—
277C22—816no—
278C22—88no—
279C22—816no—
280C22—82no—
281C22—82no—
282C22—164no—
283C22#230811yescharacteristic
284C22#231112yes—
285C22#237614yes—
286C22#281314yes—
287C22#281414yes—
288C4—48no—
289C4—42no—
290C4—84no—
291C4—82no—
292C4—168no—
293C2—22no—
294C2—24no—
295C2—24no—
296C2—24no—
297C2—84no—
298C2—82no—
299C2#1465811yescharacteristic
300C2#2177612yes—
subgroupG/H
center, socleC22#2308
commutatorC24C23×C4
Frattini, cosocleC25C24
prime radicalC22×#120C22
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. #11205C2semidirect
4. #11166C2semidirect
5. #11069C2semidirect
10. #2482C4semidirect
12. #2405C4semidirect
77. C23×C4C22⋊C4semidirect
87. C2×C4⋊C4C22⋊C4semidirect
109. C2×C22⋊C4C22⋊C4semidirect
113. C2×C42C22⋊C4semidirect

Maximal subgroups

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

subgroupindex
#353502
#206332
#112052
#111662
#110692
#109652