Finite Group Databasebeta

← #237836

Subgroups — #237836

Order 512.

1132 conjugacy classes of subgroups; only the 226 normal subgroups are shown.

normal subgroups only

This group has 1132 conjugacy classes of subgroups — too many to draw. Only its 226 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#21759C212yesmaximal
3#14159C211yesmaximal, characteristic
4#10347C214yesmaximal, complemented
5#10345C214yesmaximal, complemented
6C2×#1294C212yesmaximal
7#7907C211yesmaximal, characteristic
8C22×C22⋊Q8C2211yescharacteristic
9C22×C22⋊C8C2212yes—
10#2851—22no—
11#2646C2214yes—
12#2569—24no—
13#2442—24no—
14#2315—22no—
15C2×#380C2212yes—
16C2×C4×C22⋊C4C2211yescharacteristic
17C22×#95C2211yescharacteristic
18#1892—24no—
19#1872—24no—
20#1824C2214yes—
21#1787C2214yes—
22#1446C2212yes—
23#1445C2212yes—
24#1436—22no—
25C22⋊#95—22no—
26#1294C2218yes—
27#1280C2214yes—
28C23×Q8C2311yescharacteristic
29C24×C4—21no—
30C24×C4C2311yescharacteristic
31C23×C8—22no—
32C2×#125—22no—
33C2×C22⋊Q8—22no—
34C2×C22⋊Q8—22no—
35C2×C22⋊Q8—22no—
36C2×C22⋊Q8—24no—
37C2×C22⋊Q8Q814yes—
38C22×C4⋊C4—21no—
39C22×C4⋊C4D414yes—
40C22×C4⋊C4C2311yescharacteristic
41C22×C22⋊C4—22no—
42C22×C22⋊C4—21no—
43C22×C22⋊C4D412yes—
44C22×C42D412yes—
45C2×#103—28no—
46C2×C22⋊C8—24no—
47C2×C22⋊C8—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#380—28no—
58#380—22no—
59#380—22no—
60#380D414yes—
61#380C2314yes—
62C4×C22⋊C4—22no—
63C4×C22⋊C4—22no—
64C2×#95—22no—
65C2×#95—24no—
66C2×#95—24no—
67C2×#95—21no—
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—22no—
81C23×C4—24no—
82C23×C4—44no—
83C23×C4—42no—
84C23×C4—44no—
85C23×C4—82no—
86C23×C4—82no—
87C23×C4C2×D414yes—
88C23×C4C2×D411yescharacteristic
89C23×C4C2×D412yes—
90C23×C4C2×D412yes—
91C23×C4C2×Q812yes—
92C23×C4#4114yes—
93C23×C4#4112yes—
94C23×C4C2411yescharacteristic
95C22×C8—24no—
96C22×C8—44no—
97C22×C8—84no—
98#125—84no—
99C22⋊Q8—28no—
100C22⋊Q8—28no—
101C22⋊Q8—44no—
102C22⋊Q8—44no—
103C2×C4⋊C4—22no—
104C2×C4⋊C4—22no—
105C2×C4⋊C4—24no—
106C2×C4⋊C4—24no—
107C2×C4⋊C4—21no—
108C2×C4⋊C4—28no—
109C2×C4⋊C4—28no—
110C2×C4⋊C4—41no—
111C2×C4⋊C4—41no—
112C2×C4⋊C4—42no—
113C2×C4⋊C4—84no—
114C2×C4⋊C4C2×D418yes—
115C2×C4⋊C4C2×Q812yes—
116C2×C22⋊C4—24no—
117C2×C22⋊C4—22no—
118C2×C22⋊C4—22no—
119C2×C22⋊C4—24no—
120C2×C22⋊C4—24no—
121C2×C22⋊C4—24no—
122C2×C22⋊C4—24no—
123C2×C22⋊C4—42no—
124C2×C22⋊C4—42no—
125C2×C22⋊C4—44no—
126C2×C22⋊C4—44no—
127C2×C42—22no—
128C2×C42—22no—
129C2×C42—82no—
130#103—88no—
131C22⋊C8—24no—
132C22⋊C8—24no—
133C22⋊C8—44no—
134#95—28no—
135#95—28no—
136#95—24no—
137#95—24no—
138#95—24no—
139#95—24no—
140#95—48no—
141#95—42no—
142#95—84no—
143C24—42no—
144C24—42no—
145C24#12012yes—
146C24C22⋊Q812yes—
147C24#12412yes—
148C24#13514yes—
149C24#13714yes—
150C24C22×D411yescharacteristic
151C2×Q8—24no—
152C2×Q8—48no—
153C2×Q8—42no—
154C2×Q8—82no—
155C2×Q8—82no—
156C2×Q8—84no—
157C22×C4—24no—
158C22×C4—24no—
159C22×C4—28no—
160C22×C4—22no—
161C22×C4—22no—
162C22×C4—24no—
163C22×C4—24no—
164C22×C4—42no—
165C22×C4—44no—
166C22×C4—44no—
167C22×C4—42no—
168C22×C4—44no—
169C22×C4—44no—
170C22×C4—44no—
171C22×C4—44no—
172C22×C4—44no—
173C22×C4—42no—
174C22×C4—48no—
175C22×C4—41no—
176C22×C4—41no—
177C22×C4—42no—
178C22×C4—41no—
179C22×C4—41no—
180C22×C4—44no—
181C22×C4—48no—
182C22×C4—82no—
183C22×C4—82no—
184C22×C4—84no—
185C22×C4—84no—
186C22×C4—84no—
187C22×C4—816no—
188C22×C4—84no—
189C22×C4—84no—
190C22×C4#12011yescharacteristic
191C22×C4#12011yescharacteristic
192C22×C4C22⋊Q814yes—
193C22×C4C22⋊Q814yes—
194C22×C4C22⋊Q812yes—
195C22×C4#12414yes—
196C22×C4#12414yes—
197C22×C4#12412yes—
198C22×C4C22×D412yes—
199C22×C4C22×Q811yescharacteristic
200C22×C4C2×#4112yes—
201C22×C4C2×#4111yescharacteristic
202C2×C8—48no—
203C2×C8—44no—
204C2×C8—88no—
205C4⋊C4—24no—
206C4⋊C4—48no—
207C4⋊C4—48no—
208C4⋊C4—42no—
209C4⋊C4—44no—
210C4⋊C4—44no—
211C4⋊C4—88no—
212C4⋊C4—88no—
213C22⋊C4—44no—
214C22⋊C4—44no—
215C22⋊C4—48no—
216C22⋊C4—44no—
217C22⋊C4—44no—
218C22⋊C4—48no—
219C42—44no—
220C42—82no—
221C23—22no—
222C23—24no—
223C23—22no—
224C23—24no—
225C23—42no—
226C23—42no—
227C23—42no—
228C23—48no—
229C23—42no—
230C23—48no—
231C23#39314yes—
232C23#45214yes—
233C23#45214yes—
234C23#47918yes—
235C23#48814yes—
236C23#49114yes—
237C23C2×#12011yescharacteristic
238C23C2×C22⋊Q811yescharacteristic
239C23C2×#12411yescharacteristic
240C23C2×#13512yes—
241C23C2×#13712yes—
242Q8—48no—
243Q8—84no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—24no—
247C2×C4—44no—
248C2×C4—44no—
249C2×C4—44no—
250C2×C4—42no—
251C2×C4—44no—
252C2×C4—42no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—42no—
256C2×C4—44no—
257C2×C4—48no—
258C2×C4—44no—
259C2×C4—48no—
260C2×C4—44no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—44no—
264C2×C4—48no—
265C2×C4—44no—
266C2×C4—48no—
267C2×C4—44no—
268C2×C4—48no—
269C2×C4—82no—
270C2×C4—82no—
271C2×C4—84no—
272C2×C4—82no—
273C2×C4—816no—
274C2×C4—82no—
275C2×C4—82no—
276C2×C4—82no—
277C2×C4—84no—
278C2×C4—168no—
279C2×C4#39312yes—
280C2×C4#39312yes—
281C2×C4C2×C22⋊Q812yes—
282C2×C4C2×#12412yes—
283C8—84no—
284C22—21no—
285C22—21no—
286C22—24no—
287C22—21no—
288C22—21no—
289C22—24no—
290C22—24no—
291C22—24no—
292C22—28no—
293C22—28no—
294C22—44no—
295C22—48no—
296C22—44no—
297C22#202618yes—
298C22C2×#39311yescharacteristic
299C22C2×#45212yes—
300C22#301812yes—
301C22#302612yes—
302C22C2×#47912yes—
303C22C2×#48811yescharacteristic
304C22C2×#49111yescharacteristic
305C4—22no—
306C4—22no—
307C4—22no—
308C4—22no—
309C4—84no—
310C4—82no—
311C4—82no—
312C4—84no—
313C4—88no—
314C2—21no—
315C2—22no—
316C2—21no—
317C2—28no—
318C2#1838514yes—
319C2C2×#202611yescharacteristic
320C2#2346111yescharacteristic
321C2#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. #10347C2semidirect
5. #10345C2semidirect

Maximal subgroups

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

subgroupindex
#350202
#217592
#141592
#103472
#103452
C2×#12942
#79072