Finite Group Databasebeta

← #319484

Subgroups — #319484

Order 512.

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

normal subgroups only

This group has 858 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#33176C211yesmaximal, characteristic
2#27876C212yesmaximal
3#25813C212yesmaximal
4#21202C211yesmaximal, characteristic
5#10686C211yesmaximal, characteristic, complemented
6#10685C211yesmaximal, characteristic, complemented
7#10309C211yesmaximal, characteristic, complemented
8#10309C211yesmaximal, characteristic, complemented
9#9836C212yesmaximal, complemented
10#9834C212yesmaximal, complemented
11#8147C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C2×#500C2212yes—
14C2×#500C2212yes—
15C2×#494C2211yescharacteristic
16C2×#470—22no—
17C2×#419—22no—
18C2×C4×M16C2211yescharacteristic
19#2857C2212yes—
20#2285C2211yescharacteristic
21#2077—24no—
22#2057—24no—
23#1990C2212yes—
24#1990C2212yes—
25#1990C2212yes—
26#1990C2212yes—
27#1971—22no—
28#1960—22no—
29#1956C2211yescharacteristic
30#1956C2211yescharacteristic
31#1956C2211yescharacteristic
32#1956C2212yes—
33#1956C2211yescharacteristic
34#1919—24no—
35#1852—24no—
36#1844C2212yes—
37#1844C2211yescharacteristic
38#1834C2212yes—
39C4×C22⋊C8C2212yes—
40C2×#322C2212yes—
41#1273C2211yescharacteristic
42#1273C2211yescharacteristic
43C23×Q8—21no—
44C22×Q16—22no—
45C22×M16D412yes—
46C2×C4⋊Q8—22no—
47C2×C4⋊Q8—22no—
48C2×C4⋊Q8—22no—
49C2×C4⋊Q8—21no—
50C2×C4⋊Q8—42no—
51C2×C4⋊Q8C2312yes—
52C2×C4⋊Q8C2311yescharacteristic
53C2×C4⋊Q8C2311yescharacteristic
54C2×C22⋊Q8—22no—
55C2×#117—21no—
56C22×C4⋊C4—22no—
57C22×C42C2311yescharacteristic
58#500—24no—
59#500—24no—
60#500—42no—
61#500—42no—
62#494—42no—
63#470—48no—
64C2×#107—24no—
65C2×#107—24no—
66C2×#106—24no—
67C2×#106—22no—
68#419—48no—
69C2×#103—22no—
70C2×#103—22no—
71C2×#103—22no—
72C2×#103—22no—
73C2×#103—24no—
74C2×C22⋊C8D414yes—
75C4×M16—42no—
76C2×#97D412yes—
77C2×C4×C8C2312yes—
78C2×C4×C8C2311yescharacteristic
79C2×C4×C8C2312yes—
80C2×C4×C8C2311yescharacteristic
81#391—22no—
82#389—22no—
83#389—22no—
84#384C2312yes—
85#384C2311yescharacteristic
86#384C2311yescharacteristic
87#382—21no—
88#380—22no—
89#380—22no—
90#380—22no—
91C4×C4⋊C4—22no—
92#340—24no—
93#340—24no—
94#340—24no—
95#340—24no—
96#340—24no—
97#336D414yes—
98#322—42no—
99C22×Q8—24no—
100C22×Q8—21no—
101C22×Q8—21no—
102C22×Q8—22no—
103C22×Q8—42no—
104C22×Q8—44no—
105C22×Q8—41no—
106C22×Q8—44no—
107C23×C4—41no—
108C23×C4C2×D412yes—
109C23×C4C2×D411yescharacteristic
110C2×Q16—44no—
111C2×Q16—44no—
112C2×Q16—44no—
113C2×M16—24no—
114C2×M16—24no—
115C2×M16—42no—
116C22×C8C2×D412yes—
117C22×C8C2×D414yes—
118C22×C8C2×D412yes—
119C22×C8C2×D414yes—
120C4⋊Q8—24no—
121C4⋊Q8—21no—
122C4⋊Q8—21no—
123C4⋊Q8—22no—
124C4⋊Q8—22no—
125C4⋊Q8—22no—
126C4⋊Q8—22no—
127C4⋊Q8—22no—
128C4⋊Q8—22no—
129C4⋊Q8—44no—
130C4⋊Q8—44no—
131C4⋊Q8—42no—
132C4⋊Q8#4112yes—
133C4⋊Q8#4112yes—
134C22⋊Q8—48no—
135#117—44no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—44no—
148C2×C4⋊C4—44no—
149C2×C4⋊C4—42no—
150C2×C4⋊C4—42no—
151C2×C22⋊C4—41no—
152C2×C22⋊C4—42no—
153C2×C42—21no—
154C2×C42—21no—
155C2×C42—22no—
156C2×C42—41no—
157C2×C42C2×D412yes—
158C2×C42C2×D411yescharacteristic
159C2×C42C2411yescharacteristic
160#107—48no—
161#107—48no—
162#106—44no—
163#106—48no—
164#103—44no—
165#103—44no—
166#103—48no—
167#103—44no—
168#103—44no—
169C22⋊C8—28no—
170#97—42no—
171C4×C8—22no—
172C4×C8—22no—
173C4×C8—22no—
174C4×C8—22no—
175C4×C8—22no—
176C4×C8—22no—
177#95—42no—
178#95—42no—
179#95—42no—
180C24#12711yescharacteristic
181C2×Q8—22no—
182C2×Q8—22no—
183C2×Q8—44no—
184C2×Q8—416no—
185C2×Q8—41no—
186C2×Q8—41no—
187C2×Q8—42no—
188C2×Q8—42no—
189C2×Q8—42no—
190C2×Q8—42no—
191C2×Q8—42no—
192C2×Q8—42no—
193C2×Q8—44no—
194C2×Q8—44no—
195C2×Q8—82no—
196C2×Q8—88no—
197C2×Q8—82no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—22no—
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—24no—
204C22×C4—24no—
205C22×C4—44no—
206C22×C4—41no—
207C22×C4—41no—
208C22×C4—41no—
209C22×C4—41no—
210C22×C4—42no—
211C22×C4—42no—
212C22×C4—82no—
213C22×C4—82no—
214C22×C4#12712yes—
215C22×C4#12711yescharacteristic
216C22×C4AGL(1,ℤ/8)14yes—
217C22×C4C22×D411yescharacteristic
218C22×C4C22×D412yes—
219Q16—88no—
220M16—48no—
221C2×C8—22no—
222C2×C8—24no—
223C2×C8—22no—
224C2×C8—24no—
225C2×C8—22no—
226C2×C8—24no—
227C2×C8—22no—
228C2×C8—22no—
229C2×C8—24no—
230C2×C8—24no—
231C2×C8AGL(1,ℤ/8)14yes—
232C2×C8#13718yes—
233C4⋊C4—24no—
234C4⋊C4—24no—
235C4⋊C4—48no—
236C4⋊C4—42no—
237C4⋊C4—42no—
238C4⋊C4—42no—
239C4⋊C4—42no—
240C4⋊C4—42no—
241C4⋊C4—42no—
242C4⋊C4—42no—
243C4⋊C4—42no—
244C4⋊C4—44no—
245C4⋊C4—44no—
246C4⋊C4—44no—
247C4⋊C4—44no—
248C4⋊C4—44no—
249C4⋊C4—84no—
250C22⋊C4—82no—
251C22⋊C4—84no—
252C42—21no—
253C42—21no—
254C42—22no—
255C42—22no—
256C42—82no—
257C42C2×#4111yescharacteristic
258C42C2×#4111yescharacteristic
259C42H(5,𝔽2)12yes—
260C23—21no—
261C23—22no—
262C23—22no—
263C23—21no—
264C23#49612yes—
265C23C2×#12711yescharacteristic
266Q8—44no—
267Q8—44no—
268Q8—88no—
269Q8—82no—
270Q8—82no—
271Q8—84no—
272Q8—84no—
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—42no—
283C2×C4—42no—
284C2×C4—88no—
285C2×C4—81no—
286C2×C4—81no—
287C2×C4—81no—
288C2×C4—81no—
289C2×C4—82no—
290C2×C4—82no—
291C2×C4—81no—
292C2×C4—81no—
293C2×C4—82no—
294C2×C4—82no—
295C2×C4—82no—
296C2×C4—82no—
297C2×C4—82no—
298C2×C4—82no—
299C2×C4#49612yes—
300C2×C4#52511yescharacteristic
301C2×C4#53411yescharacteristic
302C2×C4#54612yes—
303C2×C4#54612yes—
304C2×C4C2×AGL(1,ℤ/8)12yes—
305C2×C4C2×AGL(1,ℤ/8)12yes—
306C2×C4C2×#13714yes—
307C2×C4#57814yes—
308C8—24no—
309C8—28no—
310C8—44no—
311C8—42no—
312C22—22no—
313C22—24no—
314C22—41no—
315C22—41no—
316C22—42no—
317C22#253911yescharacteristic
318C22C2×#49611yescharacteristic
319C22#315111yescharacteristic
320C22#315212yes—
321C22#315612yes—
322C4—22no—
323C4—42no—
324C4—82no—
325C4—82no—
326C4—161no—
327C4—161no—
328C4—162no—
329C4—162no—
330C4#310912yes—
331C4#319212yes—
332C4#331214yes—
333C4#332814yes—
334C2—42no—
335C2#1967512yes—
336C2#1973111yescharacteristic
337C2#1974611yescharacteristic
338C2#2831011yescharacteristic
339C2#2835512yes—
subgroupG/H
center, socleC23C2×#127
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#127
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
5. #10686C2semidirect
6. #10685C2semidirect
7. #10309C2semidirect
8. #10309C2semidirect
9. #9836C2semidirect
10. #9834C2semidirect

Maximal subgroups

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

subgroupindex
#331762
#278762
#258132
#212022
#106862
#106852
#103092
#98362
#98342
#81472