Finite Group Databasebeta

← #237816

Subgroups — #237816

Order 512.

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

normal subgroups only

This group has 962 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#35181C211yesmaximal, characteristic
2#20734C212yesmaximal
3#13925C211yesmaximal, characteristic
4#10347C214yesmaximal
5#10345C214yesmaximal
6#8026C212yesmaximal
7#7908C211yesmaximal, characteristic
8C22×C4×Q8C2211yescharacteristic
9C2×#431C2212yes—
10#2851—22no—
11#2646C2214yes—
12#2571—22no—
13#2570—24no—
14#2552—22no—
15#2317—22no—
16#2285C2212yes—
17C2×#382C2211yescharacteristic
18C2×C4×C4⋊C4C2211yescharacteristic
19#1824C2214yes—
20#1803—22no—
21#1803—22no—
22#1787C2214yes—
23C4×#103—24no—
24#1446C2212yes—
25#1445C2212yes—
26#1436—22no—
27C4⋊#95—22no—
28#1294C2218yes—
29#1280C2214yes—
30C23×Q8C2311yescharacteristic
31C2×#125—22no—
32C2×C4×Q8—22no—
33C2×C4×Q8—22no—
34C2×C4×Q8—22no—
35C2×C4×Q8—24no—
36C2×C4×Q8Q814yes—
37C2×#117—22no—
38C22×C4⋊C4—21no—
39C22×C4⋊C4—21no—
40C22×C4⋊C4D412yes—
41C22×C4⋊C4C2311yescharacteristic
42C22×C42—21no—
43C22×C42D412yes—
44C22×C42C2311yescharacteristic
45#431—24no—
46#431—24no—
47C2×C4⋊C8—22no—
48C2×#103—24no—
49C2×#103—24no—
50C2×C22⋊C8C2314yes—
51C2×C4×C8—22no—
52#400—28no—
53#398—28no—
54#389—24no—
55#389—24no—
56#389—24no—
57#387—28no—
58#385—28no—
59#384—24no—
60#384D412yes—
61#384D412yes—
62#382—24no—
63#382—22no—
64#382—22no—
65#382—24no—
66#380—22no—
67#380C2314yes—
68C4×C4⋊C4—22no—
69C4×C4⋊C4—22no—
70C4×C4⋊C4—22no—
71C4×C4⋊C4D414yes—
72C2×#95—24no—
73C2×#95—21no—
74C2×#95—24no—
75C2×#95C2312yes—
76C2×#95C2312yes—
77#340—24no—
78#340—24no—
79C22×Q8—21no—
80C22×Q8—22no—
81C22×Q8—22no—
82C22×Q8—44no—
83C22×Q8C2×Q812yes—
84C23×C4—21no—
85C23×C4—21no—
86C23×C4—22no—
87C23×C4C2×D414yes—
88C23×C4C2×D411yescharacteristic
89C23×C4C2×D411yescharacteristic
90C23×C4C2411yescharacteristic
91C22×C8—24no—
92#125—84no—
93C4×Q8—28no—
94C4×Q8—28no—
95C4×Q8—44no—
96C4×Q8—44no—
97#117—44no—
98#117—44no—
99C2×C4⋊C4—22no—
100C2×C4⋊C4—22no—
101C2×C4⋊C4—22no—
102C2×C4⋊C4—22no—
103C2×C4⋊C4—22no—
104C2×C4⋊C4—22no—
105C2×C4⋊C4—22no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4—24no—
108C2×C4⋊C4—24no—
109C2×C4⋊C4—24no—
110C2×C4⋊C4—21no—
111C2×C4⋊C4—28no—
112C2×C4⋊C4—28no—
113C2×C4⋊C4—41no—
114C2×C4⋊C4—41no—
115C2×C4⋊C4—42no—
116C2×C4⋊C4—42no—
117C2×C4⋊C4—44no—
118C2×C4⋊C4C2×D414yes—
119C2×C4⋊C4C2×D414yes—
120C2×C4⋊C4C2×Q812yes—
121C2×C22⋊C4—24no—
122C2×C42—24no—
123C2×C42—22no—
124C2×C42—24no—
125C2×C42—22no—
126C2×C42—42no—
127C2×C42—42no—
128C2×C42—82no—
129C2×C42C2×D412yes—
130C2×C42C2×D412yes—
131C2×C42C2×Q812yes—
132C2×C42#4114yes—
133C2×C42#4112yes—
134C4⋊C8—44no—
135#103—48no—
136#103—48no—
137C22⋊C8—24no—
138C22⋊C8—24no—
139C4×C8—44no—
140#95—28no—
141#95—28no—
142#95—24no—
143#95—24no—
144#95—24no—
145#95—24no—
146#95—48no—
147#95—42no—
148#95—84no—
149C24C22×D411yescharacteristic
150C2×Q8—24no—
151C2×Q8—24no—
152C2×Q8—24no—
153C2×Q8—44no—
154C2×Q8—42no—
155C2×Q8—42no—
156C2×Q8—44no—
157C2×Q8—48no—
158C2×Q8—42no—
159C22×C4—24no—
160C22×C4—22no—
161C22×C4—24no—
162C22×C4—24no—
163C22×C4—24no—
164C22×C4—22no—
165C22×C4—24no—
166C22×C4—24no—
167C22×C4—41no—
168C22×C4—41no—
169C22×C4—42no—
170C22×C4—42no—
171C22×C4—41no—
172C22×C4—41no—
173C22×C4—44no—
174C22×C4—44no—
175C22×C4—42no—
176C22×C4—42no—
177C22×C4—41no—
178C22×C4—41no—
179C22×C4—44no—
180C22×C4—42no—
181C22×C4—42no—
182C22×C4—42no—
183C22×C4—44no—
184C22×C4#12011yescharacteristic
185C22×C4#12011yescharacteristic
186C22×C4#12012yes—
187C22×C4C22⋊Q812yes—
188C22×C4C22⋊Q812yes—
189C22×C4#12412yes—
190C22×C4#12412yes—
191C22×C4#13514yes—
192C22×C4#13714yes—
193C22×C4C22×D412yes—
194C22×C4C22×Q811yescharacteristic
195C22×C4C2×#4112yes—
196C22×C4C2×#4111yescharacteristic
197C2×C8—44no—
198C2×C8—44no—
199C2×C8—44no—
200C4⋊C4—24no—
201C4⋊C4—44no—
202C4⋊C4—44no—
203C4⋊C4—44no—
204C4⋊C4—44no—
205C4⋊C4—42no—
206C4⋊C4—44no—
207C4⋊C4—44no—
208C4⋊C4—44no—
209C4⋊C4—44no—
210C4⋊C4—44no—
211C4⋊C4—44no—
212C4⋊C4—88no—
213C4⋊C4—88no—
214C4⋊C4—84no—
215C22⋊C4—44no—
216C22⋊C4—44no—
217C42—44no—
218C42—48no—
219C42—48no—
220C42—82no—
221C42C22⋊Q814yes—
222C42C22⋊Q814yes—
223C42#12414yes—
224C42#12414yes—
225C23—21no—
226C23—21no—
227C23—22no—
228C23C2×#12011yescharacteristic
229C23C2×C22⋊Q811yescharacteristic
230C23C2×#12411yescharacteristic
231C23C2×#13512yes—
232C23C2×#13712yes—
233Q8—28no—
234Q8—44no—
235Q8—44no—
236Q8—44no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—24no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—44no—
251C2×C4—42no—
252C2×C4—44no—
253C2×C4—48no—
254C2×C4—44no—
255C2×C4—42no—
256C2×C4—44no—
257C2×C4—44no—
258C2×C4—44no—
259C2×C4—44no—
260C2×C4—44no—
261C2×C4—44no—
262C2×C4—44no—
263C2×C4—44no—
264C2×C4—44no—
265C2×C4—82no—
266C2×C4—81no—
267C2×C4—81no—
268C2×C4—84no—
269C2×C4—84no—
270C2×C4—84no—
271C2×C4—82no—
272C2×C4—82no—
273C2×C4—82no—
274C2×C4#39312yes—
275C2×C4#39312yes—
276C2×C4#39314yes—
277C2×C4#45214yes—
278C2×C4#45214yes—
279C2×C4#47918yes—
280C2×C4#48814yes—
281C2×C4#49114yes—
282C2×C4C2×C22⋊Q812yes—
283C2×C4C2×#12412yes—
284C8—84no—
285C22—22no—
286C22—22no—
287C22—22no—
288C22—22no—
289C22—21no—
290C22—21no—
291C22—21no—
292C22—21no—
293C22C2×#39311yescharacteristic
294C22C2×#45212yes—
295C22#301911yescharacteristic
296C22#302011yescharacteristic
297C22#302512yes—
298C22C2×#47912yes—
299C22C2×#48811yescharacteristic
300C22C2×#49111yescharacteristic
301C4—22no—
302C4—22no—
303C4—22no—
304C4—22no—
305C4—44no—
306C4—84no—
307C4—84no—
308C4—84no—
309C4—82no—
310C4—162no—
311C4#202618yes—
312C2—21no—
313C2—22no—
314C2—21no—
315C2#1821512yes—
316C2#1822312yes—
317C2C2×#202611yescharacteristic
318C2#2346211yescharacteristic
319C2#2346711yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Maximal subgroups

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

subgroupindex
#351812
#207342
#139252
#103472
#103452
#80262
#79082