Finite Group Databasebeta

← #417005

Subgroups — #417005

Order 512.

1324 conjugacy classes of subgroups; only the 234 normal subgroups are shown.

normal subgroups only

This group has 1324 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#35020C211yesmaximal, characteristic
2#13923C211yesmaximal, characteristic
3#13923C212yesmaximal
4#11287C214yesmaximal
5#11287C214yesmaximal
6#10854C212yesmaximal
7C4×#342C211yesmaximal, characteristic
8C23×C4⋊C4C2211yescharacteristic
9#2851—22no—
10#2646—22no—
11#2569—24no—
12#2442—24no—
13#2315C2214yes—
14C2×#380C2211yescharacteristic
15C2×#380C2212yes—
16C2×C4×C22⋊C4C2212yes—
17C2×C4×C22⋊C4C2211yescharacteristic
18C2×#342C2218yes—
19C2×#342C2214yes—
20#1441—22no—
21#1441—24no—
22#1436—22no—
23#1436—24no—
24C22⋊#95C2212yes—
25C22⋊#95C2212yes—
26C22⋊#95C2214yes—
27C22⋊#95C2214yes—
28C24×C4C2311yescharacteristic
29C24×C4C2311yescharacteristic
30C24×C4C2311yescharacteristic
31C2×#125—22no—
32C22×C4⋊C4—21no—
33C22×C4⋊C4—22no—
34C22×C4⋊C4—21no—
35C22×C4⋊C4—22no—
36C22×C4⋊C4—22no—
37C22×C4⋊C4—24no—
38C22×C4⋊C4—21no—
39C22×C4⋊C4D414yes—
40C22×C4⋊C4Q814yes—
41C22×C22⋊C4C2312yes—
42C22×C22⋊C4C2312yes—
43C22×C22⋊C4C2314yes—
44C22×C22⋊C4C2314yes—
45C22×C42—22no—
46C22×C42D412yes—
47#400—28no—
48#398—28no—
49#389—24no—
50#387—28no—
51#385—28no—
52#385—28no—
53#382—22no—
54#380—22no—
55#380—28no—
56#380—22no—
57#380—22no—
58#380—22no—
59#380—24no—
60C4×C22⋊C4—22no—
61C4×C22⋊C4—22no—
62C4×C22⋊C4—24no—
63C4×C22⋊C4—24no—
64C2×#95—28no—
65C2×#95—24no—
66C2×#95—21no—
67C2×#95—22no—
68C2×#95—28no—
69C2×#95—24no—
70C2×#95D414yes—
71C2×#95D412yes—
72#342—28no—
73#342—216no—
74C25C2411yescharacteristic
75C23×C4—21no—
76C23×C4—22no—
77C23×C4—24no—
78C23×C4—21no—
79C23×C4—22no—
80C23×C4—22no—
81C23×C4—21no—
82C23×C4—22no—
83C23×C4—21no—
84C23×C4—22no—
85C23×C4—44no—
86C23×C4—82no—
87C23×C4—82no—
88C23×C4C2×D418yes—
89C23×C4C2×D414yes—
90C23×C4C2×D411yescharacteristic
91C23×C4C2×D412yes—
92C23×C4C2×D411yescharacteristic
93C23×C4C2×D412yes—
94C23×C4C2×Q812yes—
95C23×C4C2×Q812yes—
96C23×C4C2×Q812yes—
97C23×C4#4112yes—
98C23×C4#4114yes—
99#125—84no—
100C2×C4⋊C4—24no—
101C2×C4⋊C4—24no—
102C2×C4⋊C4—28no—
103C2×C4⋊C4—24no—
104C2×C4⋊C4—28no—
105C2×C4⋊C4—28no—
106C2×C4⋊C4—28no—
107C2×C4⋊C4—24no—
108C2×C4⋊C4—41no—
109C2×C4⋊C4—42no—
110C2×C4⋊C4—41no—
111C2×C4⋊C4—42no—
112C2×C4⋊C4—44no—
113C2×C4⋊C4—44no—
114C2×C4⋊C4—42no—
115C2×C4⋊C4—42no—
116C2×C4⋊C4—84no—
117C2×C4⋊C4—82no—
118C2×C22⋊C4—24no—
119C2×C22⋊C4—24no—
120C2×C22⋊C4—24no—
121C2×C22⋊C4—24no—
122C2×C22⋊C4—22no—
123C2×C22⋊C4—22no—
124C2×C22⋊C4—24no—
125C2×C22⋊C4—24no—
126C2×C22⋊C4—28no—
127C2×C22⋊C4—24no—
128C2×C22⋊C4—28no—
129C2×C22⋊C4—24no—
130C2×C22⋊C4—24no—
131C2×C22⋊C4—28no—
132C2×C22⋊C4—24no—
133C2×C22⋊C4#4118yes—
134C2×C42—22no—
135C2×C42—22no—
136C2×C42—44no—
137C2×C42—82no—
138C2×C42—84no—
139#95—28no—
140#95—28no—
141#95—24no—
142#95—24no—
143#95—24no—
144#95—48no—
145#95—416no—
146#95—44no—
147#95—42no—
148#95—84no—
149#95—88no—
150C24—42no—
151C24—42no—
152C24C22×D411yescharacteristic
153C24C22×D412yes—
154C24C22×Q811yescharacteristic
155C24C2×#4112yes—
156C24C2×#4111yescharacteristic
157C24C2×#4114yes—
158C242−1+414yes—
159C22×C4—24no—
160C22×C4—28no—
161C22×C4—22no—
162C22×C4—24no—
163C22×C4—24no—
164C22×C4—28no—
165C22×C4—22no—
166C22×C4—24no—
167C22×C4—24no—
168C22×C4—28no—
169C22×C4—24no—
170C22×C4—216no—
171C22×C4—44no—
172C22×C4—42no—
173C22×C4—44no—
174C22×C4—44no—
175C22×C4—42no—
176C22×C4—48no—
177C22×C4—44no—
178C22×C4—42no—
179C22×C4—44no—
180C22×C4—41no—
181C22×C4—41no—
182C22×C4—42no—
183C22×C4—44no—
184C22×C4—44no—
185C22×C4—48no—
186C22×C4—48no—
187C22×C4—44no—
188C22×C4—48no—
189C22×C4—41no—
190C22×C4—44no—
191C22×C4—41no—
192C22×C4—42no—
193C22×C4—44no—
194C22×C4—48no—
195C22×C4—48no—
196C22×C4—44no—
197C22×C4—44no—
198C22×C4—42no—
199C22×C4—44no—
200C22×C4—82no—
201C22×C4—82no—
202C22×C4—82no—
203C22×C4—84no—
204C22×C4—84no—
205C22×C4—82no—
206C22×C4#12011yescharacteristic
207C22×C4#12012yes—
208C22×C4#12011yescharacteristic
209C22×C4C22⋊Q812yes—
210C22×C4C22⋊Q812yes—
211C22×C4C22⋊Q814yes—
212C22×C4C22⋊Q814yes—
213C22×C4#12412yes—
214C22×C4#12414yes—
215C22×C4#12414yes—
216C22×C4#12412yes—
217C4⋊C4—48no—
218C4⋊C4—44no—
219C4⋊C4—48no—
220C4⋊C4—44no—
221C4⋊C4—84no—
222C4⋊C4—88no—
223C4⋊C4—88no—
224C4⋊C4—84no—
225C22⋊C4—28no—
226C22⋊C4—28no—
227C22⋊C4—44no—
228C22⋊C4—44no—
229C22⋊C4—44no—
230C22⋊C4—44no—
231C22⋊C4—44no—
232C22⋊C4—48no—
233C22⋊C4—44no—
234C22⋊C4—48no—
235C42—44no—
236C42—82no—
237C42—88no—
238C23—22no—
239C23—24no—
240C23—22no—
241C23—24no—
242C23—41no—
243C23—42no—
244C23—41no—
245C23—41no—
246C23—42no—
247C23—41no—
248C23—48no—
249C23—48no—
250C23C2×#12011yescharacteristic
251C23C2×C22⋊Q811yescharacteristic
252C23C2×C22⋊Q812yes—
253C23C2×#12411yescharacteristic
254C23C2×#12412yes—
255C23#52914yes—
256C23#53212yes—
257C23#53612yes—
258C23#54114yes—
259C23#54714yes—
260C23#54814yes—
261C23#55214yes—
262C23#55714yes—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—24no—
266C2×C4—24no—
267C2×C4—28no—
268C2×C4—44no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—44no—
272C2×C4—44no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—44no—
276C2×C4—42no—
277C2×C4—44no—
278C2×C4—48no—
279C2×C4—44no—
280C2×C4—42no—
281C2×C4—44no—
282C2×C4—416no—
283C2×C4—48no—
284C2×C4—44no—
285C2×C4—48no—
286C2×C4—42no—
287C2×C4—44no—
288C2×C4—48no—
289C2×C4—48no—
290C2×C4—44no—
291C2×C4—48no—
292C2×C4—44no—
293C2×C4—48no—
294C2×C4—84no—
295C2×C4—88no—
296C2×C4—84no—
297C2×C4—84no—
298C2×C4—82no—
299C2×C4—82no—
300C2×C4—84no—
301C2×C4—816no—
302C2×C4—82no—
303C2×C4—88no—
304C2×C4—82no—
305C2×C4—82no—
306C2×C4#39312yes—
307C2×C4#39314yes—
308C2×C4#39312yes—
309C22—24no—
310C22—28no—
311C22—24no—
312C22—28no—
313C22—21no—
314C22—22no—
315C22—22no—
316C22—21no—
317C22—21no—
318C22—22no—
319C22—22no—
320C22—21no—
321C22—44no—
322C22—44no—
323C22—44no—
324C22—44no—
325C22C2×#39311yescharacteristic
326C22#243212yes—
327C22#245314yes—
328C22#248914yes—
329C22#256512yes—
330C22#264211yescharacteristic
331C22#286111yescharacteristic
332C22#302512yes—
333C22#302612yes—
334C4—22no—
335C4—22no—
336C4—24no—
337C4—44no—
338C4—88no—
339C4—84no—
340C4—88no—
341C4—82no—
342C4—162no—
343C2—24no—
344C2—21no—
345C2—22no—
346C2—21no—
347C2—24no—
348C2#1523811yescharacteristic
349C2#2346711yescharacteristic
350C2#2346811yescharacteristic
351C2#2566514yes—
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, prime radical, cosocleC25C24
LCSG ▷ C25 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#350202
#139232
#112872
#108542
C4×#3422