Finite Group Databasebeta

← #241146

Subgroups — #241146

Order 512.

972 conjugacy classes of subgroups; only the 206 normal subgroups are shown.

normal subgroups only

This group has 972 conjugacy classes of subgroups — too many to draw. Only its 206 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#34578C211yesmaximal, characteristic
2#21941C212yesmaximal
3#14295C211yesmaximal, characteristic
4#10571C214yesmaximal, complemented
5#10570C212yesmaximal, complemented
6#10567C212yesmaximal, complemented
7#8024C212yesmaximal
8#7907C211yesmaximal, characteristic
9C22×C22⋊C8C2212yes—
10#2667—22no—
11#2665—22no—
12#2659C2212yes—
13#2646C2212yes—
14#2626—22no—
15#2396—24no—
16C4×C22⋊Q8—22no—
17#2280C2212yes—
18C2×#381C2211yescharacteristic
19C2×#380C2211yescharacteristic
20C2×C4×C22⋊C4C2211yescharacteristic
21#1854—24no—
22#1853C2214yes—
23#1852C2214yes—
24#1808—22no—
25#1807—22no—
26#1444C2212yes—
27#1442C2212yes—
28#1439—22no—
29#1436—22no—
30#1294C2214yes—
31#1294C2214yes—
32#1280C2211yescharacteristic
33#1280C2211yescharacteristic
34#1280C2212yes—
35C24×C4C2311yescharacteristic
36C23×C8—22no—
37C2×C22⋊Q8—22no—
38C2×C4×Q8—22no—
39C2×#117D414yes—
40C22×C4⋊C4—22no—
41C22×C4⋊C4C2312yes—
42C22×C22⋊C4—21no—
43C22×C22⋊C4—21no—
44C22×C22⋊C4D412yes—
45C22×C42D412yes—
46C2×#107—24no—
47C2×#106—24no—
48C2×C22⋊C8—22no—
49C2×C22⋊C8—24no—
50C2×C22⋊C8Q818yescomplemented
51C2×C22⋊C8C2312yes—
52C2×C22⋊C8C2312yes—
53#400—24no—
54#399—24no—
55#398—24no—
56#398—24no—
57#396—24no—
58#396—24no—
59#395—24no—
60#393—24no—
61#391—22no—
62#389—22no—
63#387—24no—
64#385—24no—
65#381—22no—
66#381—22no—
67#381Q814yescomplemented
68#380—22no—
69#380—22no—
70#380—22no—
71#380D412yes—
72#380D412yes—
73#380C2314yes—
74#379—22no—
75C4×C4⋊C4—24no—
76C4×C4⋊C4—22no—
77C4×C22⋊C4—22no—
78C4×C22⋊C4—22no—
79C4×C22⋊C4—22no—
80C4×C22⋊C4—22no—
81C2×#95—21no—
82C2×#95—21no—
83C2×#95—22no—
84C2×#95—22no—
85C2×#95—22no—
86C2×#95C2311yescharacteristic
87C2×#95C2311yescharacteristic
88C2×#95C2312yes—
89#340—24no—
90#340—24no—
91C25C2×D411yescharacteristic
92C22×Q8—22no—
93C23×C4—22no—
94C23×C4—22no—
95C23×C4—82no—
96C23×C4C2×D412yes—
97C23×C4C2×D412yes—
98C23×C4C2×D411yescharacteristic
99C23×C4C2×D412yes—
100C23×C4C2×D412yes—
101C23×C4C2×Q814yes—
102C23×C4C2×Q812yes—
103C23×C4#4112yes—
104C23×C4C2411yescharacteristic
105C22×C8—22no—
106C22×C8—22no—
107C22×C8—28no—
108C22×C8—84no—
109C22⋊Q8—44no—
110C22⋊Q8—44no—
111C4×Q8—84no—
112#117—84no—
113C2×C4⋊C4—24no—
114C2×C4⋊C4—22no—
115C2×C4⋊C4—24no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—24no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—24no—
120C2×C4⋊C4—44no—
121C2×C4⋊C4—82no—
122C2×C4⋊C4—82no—
123C2×C4⋊C4—82no—
124C2×C4⋊C4C2×D414yes—
125C2×C4⋊C4C2×D414yes—
126C2×C22⋊C4—24no—
127C2×C22⋊C4—24no—
128C2×C22⋊C4—24no—
129C2×C22⋊C4—24no—
130C2×C22⋊C4—22no—
131C2×C22⋊C4—22no—
132C2×C22⋊C4—24no—
133C2×C22⋊C4—42no—
134C2×C22⋊C4—42no—
135C2×C22⋊C4—82no—
136C2×C22⋊C4—81no—
137C2×C22⋊C4—81no—
138C2×C42—24no—
139C2×C42—22no—
140C2×C42—22no—
141C2×C42—82no—
142#107—84no—
143#106—84no—
144C22⋊C8—28no—
145C22⋊C8—22no—
146C22⋊C8—22no—
147C22⋊C8C2×Q814yes—
148C22⋊C8C2×Q814yes—
149#95—24no—
150#95—24no—
151#95—24no—
152#95—22no—
153#95—22no—
154#95—24no—
155#95—22no—
156#95—22no—
157#95—24no—
158#95—22no—
159#95—44no—
160#95—41no—
161#95—41no—
162#95—82no—
163#95—82no—
164#95C2×Q814yes—
165C24—22no—
166C24—22no—
167C24—41no—
168C24—42no—
169C24—41no—
170C24#12012yes—
171C24C22⋊Q812yes—
172C24C4⋊Q812yes—
173C24C22×D411yescharacteristic
174C2×Q8—44no—
175C2×Q8—82no—
176C2×Q8—82no—
177C22×C4—22no—
178C22×C4—22no—
179C22×C4—22no—
180C22×C4—24no—
181C22×C4—24no—
182C22×C4—24no—
183C22×C4—22no—
184C22×C4—22no—
185C22×C4—42no—
186C22×C4—42no—
187C22×C4—44no—
188C22×C4—44no—
189C22×C4—42no—
190C22×C4—42no—
191C22×C4—42no—
192C22×C4—42no—
193C22×C4—42no—
194C22×C4—42no—
195C22×C4—42no—
196C22×C4—42no—
197C22×C4—42no—
198C22×C4—42no—
199C22×C4—42no—
200C22×C4—42no—
201C22×C4—42no—
202C22×C4—42no—
203C22×C4—42no—
204C22×C4—42no—
205C22×C4—42no—
206C22×C4—84no—
207C22×C4—82no—
208C22×C4—82no—
209C22×C4#12011yescharacteristic
210C22×C4#12011yescharacteristic
211C22×C4C22⋊Q814yes—
212C22×C4C22⋊Q814yes—
213C22×C4C22⋊Q812yes—
214C22×C4C4⋊Q814yes—
215C22×C4C4⋊Q814yes—
216C22×C4C4⋊Q812yes—
217C22×C4#13514yes—
218C22×C4C22×D412yes—
219C22×C4C22×Q812yes—
220C22×C4C22×Q811yescharacteristic
221C22×C4C2×#4111yescharacteristic
222C2×C8—24no—
223C2×C8—24no—
224C2×C8—44no—
225C2×C8—44no—
226C2×C8—88no—
227C4⋊C4—44no—
228C4⋊C4—44no—
229C4⋊C4—44no—
230C4⋊C4—48no—
231C4⋊C4—44no—
232C4⋊C4—44no—
233C4⋊C4—82no—
234C4⋊C4—84no—
235C4⋊C4—82no—
236C4⋊C4—84no—
237C4⋊C4—88no—
238C22⋊C4—48no—
239C22⋊C4—44no—
240C22⋊C4—44no—
241C22⋊C4—82no—
242C22⋊C4—82no—
243C22⋊C4—84no—
244C22⋊C4—84no—
245C22⋊C4—84no—
246C42—44no—
247C42—48no—
248C42—82no—
249C23—21no—
250C23—21no—
251C23—22no—
252C23—24no—
253C23—22no—
254C23—22no—
255C23—24no—
256C23—24no—
257C23—22no—
258C23—22no—
259C23—42no—
260C23—42no—
261C23—42no—
262C23—42no—
263C23—82no—
264C23—82no—
265C23—82no—
266C23—82no—
267C23#39514yes—
268C23C2×#12011yescharacteristic
269C23C2×C22⋊Q811yescharacteristic
270C23C2×C4⋊Q811yescharacteristic
271C23C2×#13512yes—
272C23#57512yes—
273Q8—88no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—24no—
277C2×C4—24no—
278C2×C4—44no—
279C2×C4—44no—
280C2×C4—42no—
281C2×C4—44no—
282C2×C4—44no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—44no—
290C2×C4—44no—
291C2×C4—44no—
292C2×C4—44no—
293C2×C4—44no—
294C2×C4—42no—
295C2×C4—44no—
296C2×C4—44no—
297C2×C4—44no—
298C2×C4—44no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—44no—
302C2×C4—44no—
303C2×C4—82no—
304C2×C4—82no—
305C2×C4—82no—
306C2×C4—82no—
307C2×C4—82no—
308C2×C4—82no—
309C2×C4—82no—
310C2×C4—82no—
311C2×C4—82no—
312C2×C4—82no—
313C2×C4—82no—
314C2×C4—82no—
315C2×C4—82no—
316C2×C4#39512yes—
317C2×C4#39512yes—
318C2×C4C2×C22⋊Q812yes—
319C2×C4C2×C4⋊Q812yes—
320C8—44no—
321C8—44no—
322C22—21no—
323C22—21no—
324C22—21no—
325C22—21no—
326C22—22no—
327C22—22no—
328C22—22no—
329C22—22no—
330C22—24no—
331C22—24no—
332C22—24no—
333C22—44no—
334C22—44no—
335C22—84no—
336C22—82no—
337C22—82no—
338C22C2×#39511yescharacteristic
339C22#300212yes—
340C22#302111yescharacteristic
341C22#302211yescharacteristic
342C22#302612yes—
343C22#307612yes—
344C22#316011yescharacteristic
345C22#316211yescharacteristic
346C4—22no—
347C4—22no—
348C4—22no—
349C4—22no—
350C4—44no—
351C4—84no—
352C4—84no—
353C4—84no—
354C4—82no—
355C4—162no—
356C2—22no—
357C2—21no—
358C2—21no—
359C2—44no—
360C2#1829012yes—
361C2#1829912yes—
362C2#2355111yescharacteristic
363C2#2356211yescharacteristic
364C2#2356511yescharacteristic
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. #10571C2semidirect
5. #10570C2semidirect
6. #10567C2semidirect
50. C2×C22⋊C8Q8semidirect
67. #381Q8semidirect

Maximal subgroups

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

subgroupindex
#345782
#219412
#142952
#105712
#105702
#105672
#80242
#79072