Finite Group Databasebeta

← #316166

Subgroups — #316166

Order 512.

462 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 130 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#33046C211yesmaximal, characteristic
2#20768C211yesmaximal, characteristic
3#10636C212yesmaximal, complemented
4#10635C212yesmaximal, complemented
5#9770C212yesmaximal, complemented
6#9195C211yesmaximal, characteristic
7#9195C211yesmaximal, characteristic
8#9193C211yesmaximal, characteristic
9#9191C211yesmaximal, characteristic
10#8791C212yesmaximal, complemented
11#8156C211yesmaximal, characteristic
12C2×#487C2211yescharacteristic
13#2905C2211yescharacteristic
14#2905C2211yescharacteristic
15C2×#405C2211yescharacteristic
16#2628C2211yescharacteristic
17#2621C2211yescharacteristic
18#1993C2211yescharacteristic
19#1960C2211yescharacteristic
20#1935—21no—
21#1933—21no—
22#1894—21no—
23#1892—21no—
24#1838—21no—
25#1838C2212yes—
26#1837—21no—
27#1837C2212yes—
28#1835C2211yescharacteristic
29#1738—21no—
30C2×#341—21no—
31#1549C2212yes—
32#1548C2212yes—
33#1546C2212yes—
34#1546C2212yes—
35#1458C2211yescharacteristic
36#1458C2211yescharacteristic
37#1450C2212yes—
38#1286C2212yes—
39#1286C2212yes—
40#1286C2212yes—
41#1286C2212yes—
42#1281C2212yes—
43C22×M16—21no—
44C22×M16—21no—
45C22×M16D411yescharacteristic
46C23×C8D411yescharacteristic
47C2×C4⋊Q8C2311yescharacteristic
48C2×C4⋊Q8C2311yescharacteristic
49C2×C22⋊Q8—21no—
50C2×#117C2311yescharacteristic
51#487—22no—
52#487—22no—
53#431C2312yes—
54#431C2312yes—
55#424—42no—
56#424—42no—
57#423—21no—
58#423—21no—
59#423C2312yes—
60#423C2312yes—
61C2×#103—21no—
62C2×#103—21no—
63C2×#103—21no—
64C2×#103—21no—
65#405—22no—
66#405—22no—
67C2×#97D411yescharacteristic
68C2×#97C2311yescharacteristic
69C2×C4×C8D411yescharacteristic
70C2×C4×C8C2311yescharacteristic
71#400—21no—
72#399—21no—
73#398—21no—
74#398—21no—
75#397—21no—
76#397—21no—
77#395—21no—
78#393—21no—
79#391—21no—
80#389—21no—
81#382C2311yescharacteristic
82#382C2311yescharacteristic
83#380—21no—
84#341—22no—
85#341—22no—
86#341—22no—
87#341—22no—
88#340—21no—
89#340—21no—
90#340—21no—
91#340—21no—
92#333—42no—
93#332—42no—
94#332—42no—
95#332—42no—
96#324—21no—
97#324—21no—
98#323—21no—
99#323—21no—
100#322—22no—
101C82—22no—
102C22×Q8—21no—
103C22×Q8—21no—
104C23×C4C2×D411yescharacteristic
105C2×M16—22no—
106C2×M16—22no—
107C2×M16—22no—
108C2×M16—22no—
109C2×M16—41no—
110C2×M16—41no—
111C2×M16—41no—
112C2×M16—42no—
113C2×M16—42no—
114C2×M16—42no—
115C2×M16—42no—
116C22×C8—22no—
117C22×C8—21no—
118C22×C8—21no—
119C22×C8—41no—
120C22×C8—41no—
121C22×C8—41no—
122C22×C8C2×D411yescharacteristic
123C22×C8C2×D411yescharacteristic
124C22×C8C2×D411yescharacteristic
125C22×C8C2×D411yescharacteristic
126C4⋊Q8—22no—
127C4⋊Q8—22no—
128C4⋊Q8—22no—
129C4⋊Q8—22no—
130C22⋊Q8—42no—
131C22⋊Q8—42no—
132#117—22no—
133#117—22no—
134C2×C4⋊C4—21no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—21no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4—41no—
141C2×C4⋊C4—41no—
142C2×C4⋊C4#4111yescharacteristic
143C2×C4⋊C4#4111yescharacteristic
144C2×C22⋊C4—41no—
145C2×C22⋊C4—41no—
146C2×C22⋊C4#4111yescharacteristic
147C2×C22⋊C4#4111yescharacteristic
148C2×C42C2×D411yescharacteristic
149C2×C42C2411yescharacteristic
150C4⋊C8—22no—
151C4⋊C8—22no—
152C4⋊C8—22no—
153C4⋊C8—22no—
154C4⋊C8—22no—
155C4⋊C8—22no—
156C4⋊C8—42no—
157C4⋊C8—42no—
158#103—42no—
159#103—42no—
160#103—41no—
161#103—41no—
162#103—42no—
163C22⋊C8—22no—
164C22⋊C8—22no—
165#97—22no—
166#97—22no—
167C4×C8—21no—
168C4×C8—21no—
169C4×C8—22no—
170C4×C8—22no—
171C4×C8#4112yes—
172C4×C8#4112yes—
173#95—21no—
174#95—21no—
175#95—21no—
176#95—21no—
177#95—41no—
178#95—41no—
179#95—41no—
180#95—41no—
181C24#12411yescharacteristic
182C2×Q8—41no—
183C2×Q8—41no—
184C2×Q8—42no—
185C2×Q8—41no—
186C2×Q8—41no—
187C2×Q8—41no—
188C2×Q8—41no—
189C2×Q8—161no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—41no—
195C22×C4—41no—
196C22×C4—41no—
197C22×C4—41no—
198C22×C4#12411yescharacteristic
199C22×C4#12411yescharacteristic
200C22×C4#12411yescharacteristic
201C22×C4#13512yes—
202C22×C4#13512yes—
203C22×C4C22×D411yescharacteristic
204C22×C4C2×#4111yescharacteristic
205C22×C4C2×#4111yescharacteristic
206M16—44no—
207M16—42no—
208M16—42no—
209M16—42no—
210M16—42no—
211M16—82no—
212M16—82no—
213C2×C8—21no—
214C2×C8—21no—
215C2×C8—21no—
216C2×C8—21no—
217C2×C8—21no—
218C2×C8—21no—
219C2×C8—21no—
220C2×C8—21no—
221C2×C8—21no—
222C2×C8—21no—
223C2×C8—42no—
224C2×C8—42no—
225C2×C8—42no—
226C2×C8—42no—
227C2×C8—42no—
228C2×C8—42no—
229C2×C8—82no—
230C2×C8—82no—
231C2×C8#12311yescharacteristic
232C2×C8#12311yescharacteristic
233C2×C8#12312yes—
234C4⋊C4—22no—
235C4⋊C4—22no—
236C4⋊C4—41no—
237C4⋊C4—41no—
238C4⋊C4—42no—
239C4⋊C4—42no—
240C4⋊C4—42no—
241C4⋊C4—41no—
242C4⋊C4—41no—
243C4⋊C4—41no—
244C4⋊C4—41no—
245C4⋊C4—82no—
246C4⋊C4—161no—
247C22⋊C4—22no—
248C22⋊C4—22no—
249C22⋊C4—82no—
250C22⋊C4—161no—
251C42—21no—
252C42—21no—
253C42C2×#4112yes—
254C42H(5,𝔽2)12yes—
255C23—21no—
256C23—21no—
257C23—41no—
258C23—41no—
259C23#48812yes—
260C23C2×#12411yescharacteristic
261Q8—81no—
262Q8—81no—
263Q8—81no—
264Q8—81no—
265Q8—162no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—81no—
279C2×C4—81no—
280C2×C4—81no—
281C2×C4—81no—
282C2×C4—81no—
283C2×C4—81no—
284C2×C4—81no—
285C2×C4—81no—
286C2×C4—81no—
287C2×C4—81no—
288C2×C4—81no—
289C2×C4—81no—
290C2×C4#48812yes—
291C2×C4C2×#12311yescharacteristic
292C2×C4#53511yescharacteristic
293C2×C4#55312yes—
294C2×C4#55312yes—
295C2×C4C2×#13511yescharacteristic
296C2×C4C2×#13511yescharacteristic
297C2×C4#57511yescharacteristic
298C2×C4#57511yescharacteristic
299C8—21no—
300C8—21no—
301C8—22no—
302C8—41no—
303C8—41no—
304C8—82no—
305C8—82no—
306C8—82no—
307C8—82no—
308C22—21no—
309C22—21no—
310C22—22no—
311C22—42no—
312C22—41no—
313C22—41no—
314C22—81no—
315C22#250411yescharacteristic
316C22C2×#48811yescharacteristic
317C22#314211yescharacteristic
318C4—22no—
319C4—22no—
320C4—22no—
321C4—41no—
322C4—41no—
323C4—161no—
324C4—161no—
325C4—161no—
326C4—161no—
327C4#309111yescharacteristic
328C4#309111yescharacteristic
329C4#321111yescharacteristic
330C4#321111yescharacteristic
331C2—21no—
332C2—21no—
333C2—42no—
334C2#2797711yescharacteristic
335C2#2798711yescharacteristic
336C2#2798811yescharacteristic
subgroupG/H
center, socleC22#2504
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#124
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. #10636C2semidirect
4. #10635C2semidirect
5. #9770C2semidirect
10. #8791C2semidirect

Maximal subgroups

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

subgroupindex
#330462
#207682
#106362
#106352
#97702
#91952
#91932
#91912
#87912
#81562