Finite Group Databasebeta

← #237800

Subgroups — #237800

Order 512.

970 conjugacy classes of subgroups; only the 194 normal subgroups are shown.

normal subgroups only

This group has 970 conjugacy classes of subgroups — too many to draw. Only its 194 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#20740C212yesmaximal
3#13850C211yesmaximal, characteristic
4#10346C214yesmaximal
5#10345C214yesmaximal
6#8022C212yesmaximal
7#7912C211yesmaximal, characteristic
8C22×C4×Q8C2211yescharacteristic
9C2×#432C2212yes—
10#2851—22no—
11#2646C2214yes—
12#2571—22no—
13#2570—24no—
14#2552—22no—
15#2317—22no—
16#2285C2212yes—
17C2×C4×C4⋊C4C2211yescharacteristic
18C2×#376C2211yescharacteristic
19#1823C2214yes—
20#1803—22no—
21#1803—22no—
22#1787C2214yes—
23#1763—24no—
24#1446C2212yes—
25#1445C2212yes—
26#1441—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⋊C4D412yes—
40C22×C4⋊C4C2311yescharacteristic
41C22×C42—21no—
42C22×C42—21no—
43C22×C42D412yes—
44C22×C42C2311yescharacteristic
45#432—24no—
46#432—24no—
47C2×C4⋊C8—22no—
48C2×#103—24no—
49C2×#103—24no—
50C2×C22⋊C8C2314yes—
51C2×#97—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#380—22no—
66#380C2314yes—
67C4×C4⋊C4—22no—
68C4×C4⋊C4—22no—
69C4×C4⋊C4—22no—
70C4×C4⋊C4D414yes—
71#376—24no—
72C2×#95—24no—
73C2×#95—24no—
74C2×#95—21no—
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×D411yescharacteristic
88C23×C4C2×D411yescharacteristic
89C23×C4C2×D414yes—
90C23×C4C2411yescharacteristic
91C22×C8—24no—
92#125—84no—
93C4×Q8—28no—
94C4×Q8—28no—
95C4×Q8—44no—
96C4×Q8—44no—
97#117—48no—
98C2×C4⋊C4—22no—
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—24no—
107C2×C4⋊C4—21no—
108C2×C4⋊C4—28no—
109C2×C4⋊C4—28no—
110C2×C4⋊C4—44no—
111C2×C4⋊C4—42no—
112C2×C4⋊C4—42no—
113C2×C4⋊C4C2×D414yes—
114C2×C4⋊C4C2×D414yes—
115C2×C4⋊C4C2×Q812yes—
116C2×C22⋊C4—24no—
117C2×C42—24no—
118C2×C42—24no—
119C2×C42—22no—
120C2×C42—24no—
121C2×C42—22no—
122C2×C42—42no—
123C2×C42—41no—
124C2×C42—41no—
125C2×C42—42no—
126C2×C42—42no—
127C2×C42—44no—
128C2×C42C2×D412yes—
129C2×C42C2×D412yes—
130C2×C42C2×Q812yes—
131C2×C42#4114yes—
132C2×C42#4112yes—
133C4⋊C8—44no—
134#103—48no—
135#103—48no—
136C22⋊C8—24no—
137C22⋊C8—24no—
138#97—44no—
139#95—28no—
140#95—28no—
141#95—24no—
142#95—24no—
143#95—24no—
144#95—24no—
145#95—48no—
146#95—48no—
147#95—42no—
148C24C22×D411yescharacteristic
149C2×Q8—24no—
150C2×Q8—24no—
151C2×Q8—24no—
152C2×Q8—44no—
153C2×Q8—42no—
154C2×Q8—42no—
155C2×Q8—44no—
156C2×Q8—48no—
157C2×Q8—42no—
158C22×C4—24no—
159C22×C4—22no—
160C22×C4—22no—
161C22×C4—24no—
162C22×C4—22no—
163C22×C4—24no—
164C22×C4—24no—
165C22×C4—24no—
166C22×C4—24no—
167C22×C4—22no—
168C22×C4—24no—
169C22×C4—24no—
170C22×C4—42no—
171C22×C4—42no—
172C22×C4—41no—
173C22×C4—41no—
174C22×C4—44no—
175C22×C4—44no—
176C22×C4—44no—
177C22×C4—42no—
178C22×C4—41no—
179C22×C4—42no—
180C22×C4—41no—
181C22×C4—44no—
182C22×C4—44no—
183C22×C4#12011yescharacteristic
184C22×C4#12012yes—
185C22×C4#12011yescharacteristic
186C22×C4C22⋊Q812yes—
187C22×C4C22⋊Q812yes—
188C22×C4#12412yes—
189C22×C4#12412yes—
190C22×C4C22×D412yes—
191C22×C4C22×Q811yescharacteristic
192C22×C4C2×#4112yes—
193C22×C4C2×#4111yescharacteristic
194C2×C8—44no—
195C2×C8—44no—
196C2×C8—44no—
197C4⋊C4—24no—
198C4⋊C4—44no—
199C4⋊C4—44no—
200C4⋊C4—44no—
201C4⋊C4—44no—
202C4⋊C4—44no—
203C4⋊C4—44no—
204C4⋊C4—42no—
205C4⋊C4—48no—
206C4⋊C4—44no—
207C4⋊C4—44no—
208C4⋊C4—88no—
209C4⋊C4—88no—
210C22⋊C4—48no—
211C42—48no—
212C42—48no—
213C42—44no—
214C42—44no—
215C42—44no—
216C42—82no—
217C42C22⋊Q814yes—
218C42C22⋊Q814yes—
219C42#12414yes—
220C42#12414yes—
221C23—21no—
222C23—21no—
223C23—22no—
224C23C2×#12011yescharacteristic
225C23C2×C22⋊Q811yescharacteristic
226C23C2×#12411yescharacteristic
227C23#57514yes—
228Q8—28no—
229Q8—44no—
230Q8—44no—
231Q8—44no—
232C2×C4—24no—
233C2×C4—24no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—24no—
241C2×C4—24no—
242C2×C4—24no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—44no—
248C2×C4—42no—
249C2×C4—44no—
250C2×C4—44no—
251C2×C4—48no—
252C2×C4—48no—
253C2×C4—44no—
254C2×C4—42no—
255C2×C4—44no—
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—84no—
265C2×C4—82no—
266C2×C4—82no—
267C2×C4—82no—
268C2×C4—82no—
269C2×C4—84no—
270C2×C4#39312yes—
271C2×C4#39314yes—
272C2×C4#39312yes—
273C2×C4#45512yes—
274C2×C4#45612yes—
275C2×C4C2×C22⋊Q812yes—
276C2×C4C2×#12412yes—
277C8—84no—
278C22—24no—
279C22—24no—
280C22—21no—
281C22—21no—
282C22—21no—
283C22—21no—
284C22C2×#39311yescharacteristic
285C22#300512yes—
286C22C2×#45511yescharacteristic
287C22C2×#45611yescharacteristic
288C22#302512yes—
289C22#308012yes—
290C22#314212yes—
291C4—24no—
292C4—22no—
293C4—22no—
294C4—22no—
295C4—44no—
296C4—84no—
297C4—84no—
298C4—84no—
299C4—84no—
300C4—82no—
301C4#202814yes—
302C2—22no—
303C2—22no—
304C2#1818312yes—
305C2#1819112yes—
306C2#2345311yescharacteristic
307C2C2×#202811yescharacteristic
308C2#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
#207402
#138502
#103462
#103452
#80222
#79122