Finite Group Databasebeta

← #331191

Subgroups — #331191

Order 512.

801 conjugacy classes of subgroups; only the 178 normal subgroups are shown.

normal subgroups only

This group has 801 conjugacy classes of subgroups — too many to draw. Only its 178 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#33172C211yesmaximal, characteristic, complemented
2#33170C211yesmaximal, characteristic
3#29876C212yesmaximal, complemented
4#29842C212yesmaximal
5#10675C212yesmaximal, complemented
6#10674C211yesmaximal, characteristic, complemented
7#10671C211yesmaximal, characteristic, complemented
8#10309C211yesmaximal, characteristic, complemented
9#10308C212yesmaximal, complemented
10#10306C211yesmaximal, characteristic, complemented
11#7849C211yesmaximal, characteristic, complemented
12C2×#500C2212yes—
13C2×#494C2211yescharacteristic
14C2×#492C2211yescharacteristic
15C2×#486C2212yes—
16C2×#467—22no—
17C2×#465—22no—
18C2×#431C2212yes—
19C22×C4×C8C2211yescharacteristic
20#2857C2212yes—
21#2857C2211yescharacteristic
22#2856C2211yescharacteristic
23#2086—24no—
24#2075—24no—
25#2067—24no—
26#2033—24no—
27#1990C2212yes—
28#1990C2212yes—
29#1990C2212yes—
30#1969—22no—
31#1968—22no—
32#1957C2211yescharacteristic
33#1957C2212yes—
34#1956C2212yes—
35#1956C2211yescharacteristic
36#1845C2212yes—
37#1845C2211yescharacteristic
38#1844C2211yescharacteristic
39#1834C2212yes—
40#1273C2211yescharacteristic
41#1273C2212yes—
42#1273C2211yescharacteristic
43C22×Q16—22no—
44C22×SD16—22no—
45C23×C8D412yes—
46C2×C4⋊Q8C2312yes—
47C2×C4⋊Q8C2311yescharacteristic
48C2×#127C2311yescharacteristic
49C2×C22⋊Q8—21no—
50C2×C22⋊Q8—22no—
51C2×#121—21no—
52C2×#121—22no—
53C22×C42C2311yescharacteristic
54#500—42no—
55#500—42no—
56#494—42no—
57#492—42no—
58#486—42no—
59#486—42no—
60#467—48no—
61#465—48no—
62#431—42no—
63#431—42no—
64C2×#107—24no—
65C2×#107—22no—
66C2×#107—24no—
67C2×#106—22no—
68C2×C4⋊C8D414yes—
69C2×#103—22no—
70C2×#103—24no—
71C2×#102—24no—
72C2×#102—22no—
73C2×#102—24no—
74C2×C22⋊C8D414yes—
75C2×C4×C8—42no—
76C2×C4×C8D412yes—
77C2×C4×C8C2312yes—
78C2×C4×C8C2311yescharacteristic
79C2×C4×C8C2311yescharacteristic
80C2×C4×C8C2312yes—
81#391—21no—
82#391—22no—
83#390—22no—
84#390—21no—
85#389—22no—
86#389—22no—
87#389—22no—
88#384C2311yescharacteristic
89#384C2311yescharacteristic
90#384C2312yes—
91#380—22no—
92#380—22no—
93#380—22no—
94#340—24no—
95#340—24no—
96#340—24no—
97#340—24no—
98C22×Q8—21no—
99C22×Q8—22no—
100C22×D4—21no—
101C22×D4—22no—
102C22×D4—41no—
103C23×C4C2×D412yes—
104C23×C4C2×D411yescharacteristic
105C2×Q16—44no—
106C2×Q16—44no—
107C2×Q16—44no—
108C2×SD16—44no—
109C2×SD16—44no—
110C2×SD16—44no—
111C22×C8—42no—
112C22×C8—42no—
113C22×C8D818yes—
114C22×C8C2×D414yes—
115C22×C8C2×D414yes—
116C22×C8C2×D412yes—
117C22×C8C2×D412yes—
118C4⋊Q8—22no—
119C4⋊Q8—22no—
120C4⋊Q8—22no—
121C4⋊Q8—22no—
122C4⋊Q8—21no—
123C4⋊Q8—22no—
124C4⋊Q8—21no—
125#127—21no—
126#127—22no—
127#127—21no—
128C22⋊Q8—44no—
129C22⋊Q8—48no—
130#121—44no—
131#121—84no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—21no—
134C2×C4⋊C4—21no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—41no—
143C2×C4⋊C4—42no—
144C2×C22⋊C4—41no—
145C2×C22⋊C4—42no—
146C2×C22⋊C4—42no—
147C2×C22⋊C4—41no—
148C2×C42—22no—
149C2×C42—21no—
150C2×C42—21no—
151C2×C42C2×D412yes—
152C2×C42C2×D411yescharacteristic
153C2×C42C2411yescharacteristic
154#107—48no—
155#107—44no—
156#107—84no—
157#106—44no—
158C4⋊C8—44no—
159#103—44no—
160#103—48no—
161#102—48no—
162#102—44no—
163#102—84no—
164C22⋊C8—44no—
165C4×C8—22no—
166C4×C8—22no—
167C4×C8—22no—
168C4×C8—22no—
169C4×C8—22no—
170C4×C8—22no—
171C4×C8—42no—
172#95—42no—
173#95—42no—
174#95—42no—
175C24—41no—
176C24#12711yescharacteristic
177C2×Q8—41no—
178C2×Q8—42no—
179C2×Q8—42no—
180C2×Q8—41no—
181C2×Q8—42no—
182C2×Q8—44no—
183C2×Q8—42no—
184C2×Q8—44no—
185C2×D4—42no—
186C2×D4—41no—
187C2×D4—41no—
188C2×D4—42no—
189C2×D4—42no—
190C2×D4—42no—
191C2×D4—44no—
192C2×D4—44no—
193C2×D4—84no—
194C2×D4—82no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—24no—
201C22×C4—24no—
202C22×C4—42no—
203C22×C4—41no—
204C22×C4—42no—
205C22×C4—41no—
206C22×C4—41no—
207C22×C4#12712yes—
208C22×C4#12711yescharacteristic
209C22×C4C2×D814yes—
210C22×C4C22×D411yescharacteristic
211C22×C4C22×D412yes—
212Q16—88no—
213SD16—88no—
214C2×C8—22no—
215C2×C8—22no—
216C2×C8—22no—
217C2×C8—22no—
218C2×C8—24no—
219C2×C8—24no—
220C2×C8—24no—
221C2×C8—24no—
222C2×C8—24no—
223C2×C8—24no—
224C2×C8—48no—
225C2×C8C2×D814yes—
226C2×C8C2×D814yes—
227C4⋊C4—42no—
228C4⋊C4—42no—
229C4⋊C4—42no—
230C4⋊C4—42no—
231C4⋊C4—42no—
232C4⋊C4—42no—
233C4⋊C4—42no—
234C4⋊C4—42no—
235C4⋊C4—44no—
236C4⋊C4—44no—
237C4⋊C4—44no—
238C4⋊C4—44no—
239C4⋊C4—44no—
240C4⋊C4—82no—
241C4⋊C4—84no—
242C22⋊C4—84no—
243C22⋊C4—82no—
244C22⋊C4—84no—
245C22⋊C4—82no—
246C42—21no—
247C42—22no—
248C42—22no—
249C42—21no—
250C42H(5,𝔽2)12yes—
251C42H(5,𝔽2)11yescharacteristic
252C422−1+411yescharacteristic
253C23—21no—
254C23—22no—
255C23—22no—
256C23—21no—
257C23—81no—
258C23—81no—
259C23—82no—
260C23—82no—
261C23—81no—
262C23#49312yes—
263C23C2×#12711yescharacteristic
264Q8—82no—
265Q8—84no—
266Q8—82no—
267Q8—84no—
268D4—82no—
269D4—84no—
270D4—82no—
271D4—84no—
272D4—164no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—24no—
276C2×C4—24no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—44no—
281C2×C4—44no—
282C2×C4—82no—
283C2×C4—81no—
284C2×C4—81no—
285C2×C4—81no—
286C2×C4—82no—
287C2×C4—81no—
288C2×C4—82no—
289C2×C4—82no—
290C2×C4—82no—
291C2×C4—82no—
292C2×C4—82no—
293C2×C4—82no—
294C2×C4—81no—
295C2×C4#49312yes—
296C2×C4#53411yescharacteristic
297C2×C4#53412yes—
298C2×C4#53711yescharacteristic
299C2×C4#53712yes—
300C2×C4C22×D812yes—
301C2×C4#57512yes—
302C2×C4#57714yes—
303C2×C4#57814yes—
304C8—24no—
305C8—24no—
306C8—44no—
307C8—44no—
308C22—22no—
309C22—22no—
310C22—22no—
311C22—42no—
312C22—42no—
313C22—161no—
314C22—162no—
315C22—161no—
316C22—161no—
317C22—162no—
318C22#265611yescharacteristic
319C22C2×#49311yescharacteristic
320C22#315011yescharacteristic
321C22#315412yes—
322C22#315612yes—
323C4—22no—
324C4—42no—
325C4—161no—
326C4—162no—
327C4—162no—
328C4—161no—
329C4—161no—
330C4#318612yes—
331C4#320114yes—
332C4#324512yes—
333C4#325814yes—
334C2—42no—
335C2—321no—
336C2#1966011yescharacteristic
337C2#1966211yescharacteristic
338C2#1970712yes—
339C2#3023911yescharacteristic
340C2#3025212yes—
subgroupG/H
center, socleC23C2×#127
commutator, Frattini, cosocleC2×C42C24
prime radical#2856C22
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #33172C2semidirect
3. #29876C2semidirect
5. #10675C2semidirect
6. #10674C2semidirect
7. #10671C2semidirect
8. #10309C2semidirect
9. #10308C2semidirect
10. #10306C2semidirect
11. #7849C2semidirect

Maximal subgroups

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

subgroupindex
#331722
#331702
#298762
#298422
#106752
#106742
#106712
#103092
#103082
#103062
#78492