Finite Group Databasebeta

← #237822

Subgroups — #237822

Order 512.

1018 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 1018 conjugacy classes of subgroups — too many to draw. Only its 186 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#35020C211yesmaximal, characteristic
2#21765C212yesmaximal
3#14292C211yesmaximal, characteristic
4#10346C214yesmaximal, complemented
5#10345C214yesmaximal, complemented
6#8015C212yesmaximal
7#7911C211yesmaximal, characteristic
8C22×C22⋊Q8C2211yescharacteristic
9C2×#407C2212yes—
10#2851—22no—
11#2646C2214yes—
12#2569—24no—
13#2442—24no—
14#2315—22no—
15C2×#381C2211yescharacteristic
16C2×#380C2212yes—
17C2×C4×C22⋊C4C2211yescharacteristic
18#1894—24no—
19#1872—24no—
20#1823C2214yes—
21#1787C2214yes—
22#1446C2212yes—
23#1445C2212yes—
24#1440—22no—
25#1436—22no—
26#1294C2218yes—
27#1280C2214yes—
28C23×Q8C2311yescharacteristic
29C24×C4C2311yescharacteristic
30C22×M16—22no—
31C2×#125—22no—
32C2×C22⋊Q8—22no—
33C2×C22⋊Q8—22no—
34C2×C22⋊Q8—22no—
35C2×C22⋊Q8—24no—
36C2×C22⋊Q8Q814yes—
37C22×C4⋊C4—21no—
38C22×C4⋊C4D414yes—
39C22×C4⋊C4C2311yescharacteristic
40C22×C22⋊C4—22no—
41C22×C22⋊C4—21no—
42C22×C22⋊C4—21no—
43C22×C22⋊C4D412yes—
44C22×C42D412yes—
45C2×#103—28no—
46#407—24no—
47#407—24no—
48C2×C22⋊C8—22no—
49C2×C22⋊C8C2314yes—
50#400—28no—
51#398—28no—
52#389—24no—
53#387—28no—
54#385—28no—
55#385—28no—
56#382—22no—
57#381—24no—
58#380—28no—
59#380—22no—
60#380—22no—
61#380D414yes—
62#380C2314yes—
63C4×C22⋊C4—22no—
64C4×C22⋊C4—22no—
65C2×#95—22no—
66C2×#95—21no—
67C2×#95—24no—
68C2×#95—24no—
69C2×#95C2312yes—
70C2×#95C2312yes—
71#340—28no—
72C25C2×D411yescharacteristic
73C22×Q8—21no—
74C22×Q8—22no—
75C22×Q8—22no—
76C22×Q8—82no—
77C22×Q8C2×Q812yes—
78C23×C4—21no—
79C23×C4—21no—
80C23×C4—24no—
81C23×C4—22no—
82C23×C4—82no—
83C23×C4C2×D414yes—
84C23×C4C2×D411yescharacteristic
85C23×C4C2×D412yes—
86C23×C4C2×D412yes—
87C23×C4C2×Q812yes—
88C23×C4#4114yes—
89C23×C4#4112yes—
90C23×C4C2411yescharacteristic
91C2×M16—44no—
92C2×M16—84no—
93C22×C8—24no—
94#125—84no—
95C22⋊Q8—28no—
96C22⋊Q8—28no—
97C22⋊Q8—44no—
98C22⋊Q8—44no—
99C2×C4⋊C4—22no—
100C2×C4⋊C4—22no—
101C2×C4⋊C4—24no—
102C2×C4⋊C4—24no—
103C2×C4⋊C4—21no—
104C2×C4⋊C4—28no—
105C2×C4⋊C4—28no—
106C2×C4⋊C4—41no—
107C2×C4⋊C4—41no—
108C2×C4⋊C4—42no—
109C2×C4⋊C4—84no—
110C2×C4⋊C4C2×D418yes—
111C2×C4⋊C4C2×Q812yes—
112C2×C22⋊C4—24no—
113C2×C22⋊C4—22no—
114C2×C22⋊C4—22no—
115C2×C22⋊C4—24no—
116C2×C22⋊C4—24no—
117C2×C22⋊C4—44no—
118C2×C22⋊C4—48no—
119C2×C22⋊C4—42no—
120C2×C22⋊C4—42no—
121C2×C22⋊C4—42no—
122C2×C22⋊C4—82no—
123C2×C42—22no—
124C2×C42—22no—
125C2×C42—82no—
126#103—88no—
127C22⋊C8—24no—
128C22⋊C8—24no—
129C22⋊C8—44no—
130#95—28no—
131#95—28no—
132#95—24no—
133#95—24no—
134#95—24no—
135#95—24no—
136#95—48no—
137#95—42no—
138#95—84no—
139C24—24no—
140C24—42no—
141C24—42no—
142C24#12012yes—
143C24C22⋊Q812yes—
144C24#12412yes—
145C24C22×D411yescharacteristic
146C2×Q8—24no—
147C2×Q8—48no—
148C2×Q8—42no—
149C2×Q8—82no—
150C2×Q8—82no—
151C2×Q8—84no—
152C22×C4—24no—
153C22×C4—28no—
154C22×C4—22no—
155C22×C4—22no—
156C22×C4—24no—
157C22×C4—24no—
158C22×C4—42no—
159C22×C4—44no—
160C22×C4—44no—
161C22×C4—42no—
162C22×C4—44no—
163C22×C4—44no—
164C22×C4—44no—
165C22×C4—41no—
166C22×C4—41no—
167C22×C4—41no—
168C22×C4—41no—
169C22×C4—44no—
170C22×C4—48no—
171C22×C4—44no—
172C22×C4—42no—
173C22×C4—42no—
174C22×C4—42no—
175C22×C4—44no—
176C22×C4—82no—
177C22×C4—82no—
178C22×C4—84no—
179C22×C4—84no—
180C22×C4—84no—
181C22×C4#12011yescharacteristic
182C22×C4#12011yescharacteristic
183C22×C4C22⋊Q814yes—
184C22×C4C22⋊Q814yes—
185C22×C4C22⋊Q812yes—
186C22×C4#12414yes—
187C22×C4#12414yes—
188C22×C4#12412yes—
189C22×C4C22×D412yes—
190C22×C4C22×Q811yescharacteristic
191C22×C4C2×#4112yes—
192C22×C4C2×#4111yescharacteristic
193M16—88no—
194C2×C8—48no—
195C2×C8—44no—
196C4⋊C4—24no—
197C4⋊C4—48no—
198C4⋊C4—48no—
199C4⋊C4—42no—
200C4⋊C4—44no—
201C4⋊C4—44no—
202C4⋊C4—88no—
203C4⋊C4—88no—
204C22⋊C4—44no—
205C22⋊C4—44no—
206C22⋊C4—48no—
207C22⋊C4—48no—
208C22⋊C4—48no—
209C22⋊C4—84no—
210C42—44no—
211C42—82no—
212C23—22no—
213C23—22no—
214C23—22no—
215C23—22no—
216C23—24no—
217C23—24no—
218C23—24no—
219C23—24no—
220C23—42no—
221C23—42no—
222C23—42no—
223C23—42no—
224C23—84no—
225C23—84no—
226C23#39314yes—
227C23C2×#12011yescharacteristic
228C23C2×C22⋊Q811yescharacteristic
229C23C2×#12411yescharacteristic
230C23#57514yes—
231Q8—48no—
232Q8—84no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—24no—
236C2×C4—44no—
237C2×C4—44no—
238C2×C4—44no—
239C2×C4—42no—
240C2×C4—44no—
241C2×C4—42no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—42no—
245C2×C4—48no—
246C2×C4—44no—
247C2×C4—44no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—44no—
251C2×C4—48no—
252C2×C4—44no—
253C2×C4—48no—
254C2×C4—44no—
255C2×C4—48no—
256C2×C4—82no—
257C2×C4—82no—
258C2×C4—82no—
259C2×C4—82no—
260C2×C4—84no—
261C2×C4—82no—
262C2×C4—82no—
263C2×C4—84no—
264C2×C4—84no—
265C2×C4—82no—
266C2×C4—168no—
267C2×C4#39312yes—
268C2×C4#39312yes—
269C2×C4C2×C22⋊Q812yes—
270C2×C4C2×#12412yes—
271C8—84no—
272C22—21no—
273C22—21no—
274C22—21no—
275C22—21no—
276C22—24no—
277C22—24no—
278C22—24no—
279C22—24no—
280C22—24no—
281C22—44no—
282C22—44no—
283C22—84no—
284C22—84no—
285C22C2×#39311yescharacteristic
286C22#300512yes—
287C22#302212yes—
288C22#302612yes—
289C22#308012yes—
290C22#314212yes—
291C4—22no—
292C4—22no—
293C4—22no—
294C4—22no—
295C4—82no—
296C4—82no—
297C4—84no—
298C4—88no—
299C4—162no—
300C2—22no—
301C2—22no—
302C2—44no—
303C2#1835514yes—
304C2#2345311yescharacteristic
305C2#2346411yescharacteristic
306C2#2346811yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #10346C2semidirect
5. #10345C2semidirect

Maximal subgroups

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

subgroupindex
#350202
#217652
#142922
#103462
#103452
#80152
#79112