Finite Group Databasebeta

← #319399

Subgroups — #319399

Order 512.

864 conjugacy classes of subgroups; only the 190 normal subgroups are shown.

normal subgroups only

This group has 864 conjugacy classes of subgroups — too many to draw. Only its 190 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#27877C212yesmaximal
3#25813C212yesmaximal
4#21202C211yesmaximal, characteristic
5#10675C212yesmaximal, complemented
6#10309C212yesmaximal, complemented
7#9831C212yesmaximal, complemented
8#9823C212yesmaximal, complemented
9#8146C211yesmaximal, characteristic
10C22×C4⋊Q8C2211yescharacteristic
11C2×#500C2212yes—
12C2×#500C2212yes—
13C2×#494C2211yescharacteristic
14C2×#470—22no—
15C2×#419—22no—
16C2×C4×M16C2211yescharacteristic
17#2857C2212yes—
18#2285C2211yescharacteristic
19#2075—24no—
20#2057—24no—
21#1990C2212yes—
22#1990C2212yes—
23#1990C2212yes—
24#1990C2212yes—
25#1971—22no—
26#1960—22no—
27#1956C2212yes—
28#1956C2212yes—
29#1956C2212yes—
30#1918—24no—
31#1852—24no—
32#1845C2212yes—
33#1844C2211yescharacteristic
34#1834C2212yes—
35C4×C22⋊C8C2212yes—
36C2×#322C2211yescharacteristic
37C2×C82C2211yescharacteristic
38#1273C2212yes—
39C23×Q8—21no—
40C22×Q16—22no—
41C22×M16D412yes—
42C2×C4⋊Q8—24no—
43C2×C4⋊Q8—22no—
44C2×C4⋊Q8—21no—
45C2×C4⋊Q8—41no—
46C2×C4⋊Q8—41no—
47C2×C4⋊Q8C2312yes—
48C2×C4⋊Q8C2312yes—
49C2×C22⋊Q8—22no—
50C2×#117—21no—
51C22×C4⋊C4—22no—
52C22×C42C2311yescharacteristic
53#500—24no—
54#500—24no—
55#500—42no—
56#500—42no—
57#494—41no—
58#494—41no—
59#470—48no—
60C2×#107—24no—
61C2×#107—24no—
62C2×#107—24no—
63C2×#106—22no—
64#419—48no—
65C2×#103—24no—
66C2×#103—24no—
67C2×#103—24no—
68C2×C22⋊C8D414yes—
69C4×M16—41no—
70C4×M16—41no—
71C2×#97D412yes—
72C2×C4×C8C2312yes—
73C2×C4×C8C2311yescharacteristic
74C2×C4×C8C2312yes—
75C2×C4×C8C2311yescharacteristic
76#391—22no—
77#389—22no—
78#389—22no—
79#384C2312yes—
80#384C2312yes—
81#382—21no—
82#380—22no—
83#380—22no—
84#380—22no—
85C4×C4⋊C4—22no—
86#340—24no—
87#340—24no—
88#340—24no—
89#340—24no—
90#340—24no—
91#336D414yes—
92#322—41no—
93C82—41no—
94C22×Q8—24no—
95C22×Q8—22no—
96C22×Q8—22no—
97C22×Q8—42no—
98C22×Q8—44no—
99C22×Q8—41no—
100C22×Q8—44no—
101C23×C4—41no—
102C23×C4C2×D412yes—
103C23×C4C2×D411yescharacteristic
104C2×Q16—44no—
105C2×Q16—44no—
106C2×Q16—44no—
107C2×M16—42no—
108C2×M16—42no—
109C2×M16D818yes—
110C22×C8C2×D414yes—
111C22×C8C2×D412yes—
112C22×C8C2×D412yes—
113C22×C8C2×D414yes—
114C4⋊Q8—24no—
115C4⋊Q8—22no—
116C4⋊Q8—22no—
117C4⋊Q8—22no—
118C4⋊Q8—22no—
119C4⋊Q8—22no—
120C4⋊Q8—22no—
121C4⋊Q8—22no—
122C4⋊Q8—44no—
123C4⋊Q8—44no—
124C4⋊Q8—42no—
125C4⋊Q8#4114yes—
126C22⋊Q8—48no—
127#117—44no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—44no—
139C2×C4⋊C4—44no—
140C2×C4⋊C4—42no—
141C2×C4⋊C4—42no—
142C2×C22⋊C4—41no—
143C2×C22⋊C4—42no—
144C2×C42—22no—
145C2×C42—21no—
146C2×C42—21no—
147C2×C42—41no—
148C2×C42C2×D412yes—
149C2×C42C2×D411yescharacteristic
150C2×C42C2411yescharacteristic
151#107—48no—
152#107—48no—
153#107—48no—
154#106—44no—
155#103—48no—
156#103—48no—
157#103—48no—
158C22⋊C8—28no—
159#97—42no—
160C4×C8—22no—
161C4×C8—22no—
162C4×C8—21no—
163C4×C8—22no—
164C4×C8—21no—
165C4×C8—22no—
166C4×C8—21no—
167C4×C8—21no—
168#95—42no—
169#95—42no—
170#95—42no—
171C24#12711yescharacteristic
172C2×Q8—24no—
173C2×Q8—44no—
174C2×Q8—416no—
175C2×Q8—42no—
176C2×Q8—42no—
177C2×Q8—42no—
178C2×Q8—44no—
179C2×Q8—44no—
180C2×Q8—44no—
181C2×Q8—44no—
182C2×Q8—82no—
183C2×Q8—88no—
184C2×Q8—82no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—22no—
188C22×C4—22no—
189C22×C4—22no—
190C22×C4—24no—
191C22×C4—24no—
192C22×C4—44no—
193C22×C4—42no—
194C22×C4—42no—
195C22×C4—42no—
196C22×C4—42no—
197C22×C4—82no—
198C22×C4—82no—
199C22×C4#12712yes—
200C22×C4#12711yescharacteristic
201C22×C4C2×D814yes—
202C22×C4C22×D411yescharacteristic
203C22×C4C22×D412yes—
204Q16—88no—
205M16—48no—
206C2×C8—22no—
207C2×C8—22no—
208C2×C8—24no—
209C2×C8—22no—
210C2×C8—22no—
211C2×C8—24no—
212C2×C8—24no—
213C2×C8—24no—
214C2×C8—24no—
215C2×C8C2×D814yes—
216C2×C8C2×D814yes—
217C2×C8#13718yes—
218C4⋊C4—24no—
219C4⋊C4—24no—
220C4⋊C4—48no—
221C4⋊C4—42no—
222C4⋊C4—42no—
223C4⋊C4—42no—
224C4⋊C4—42no—
225C4⋊C4—42no—
226C4⋊C4—42no—
227C4⋊C4—44no—
228C4⋊C4—44no—
229C4⋊C4—44no—
230C4⋊C4—44no—
231C4⋊C4—44no—
232C4⋊C4—44no—
233C4⋊C4—84no—
234C22⋊C4—82no—
235C22⋊C4—84no—
236C42—21no—
237C42—21no—
238C42—22no—
239C42—22no—
240C42—82no—
241C42C2×#4112yes—
242C42H(5,𝔽2)11yescharacteristic
243C42H(5,𝔽2)11yescharacteristic
244C23—22no—
245C23—21no—
246C23—22no—
247C23—21no—
248C23#49312yes—
249C23C2×#12711yescharacteristic
250Q8—48no—
251Q8—88no—
252Q8—84no—
253Q8—84no—
254Q8—84no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—24no—
258C2×C4—24no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—44no—
263C2×C4—44no—
264C2×C4—44no—
265C2×C4—88no—
266C2×C4—82no—
267C2×C4—82no—
268C2×C4—82no—
269C2×C4—82no—
270C2×C4—82no—
271C2×C4—82no—
272C2×C4—82no—
273C2×C4—82no—
274C2×C4—82no—
275C2×C4—82no—
276C2×C4—82no—
277C2×C4#49312yes—
278C2×C4#52511yescharacteristic
279C2×C4#53411yescharacteristic
280C2×C4#54612yes—
281C2×C4#54612yes—
282C2×C4C22×D812yes—
283C2×C4C2×#13714yes—
284C2×C4#57512yes—
285C2×C4#57814yes—
286C8—24no—
287C8—24no—
288C8—28no—
289C8—44no—
290C22—22no—
291C22—24no—
292C22—41no—
293C22—41no—
294C22—42no—
295C22#253911yescharacteristic
296C22C2×#49311yescharacteristic
297C22#315212yes—
298C22#315311yescharacteristic
299C22#315612yes—
300C4—22no—
301C4—42no—
302C4—84no—
303C4—162no—
304C4—162no—
305C4—162no—
306C4#310012yes—
307C4#318612yes—
308C4#331214yes—
309C4#332814yes—
310C2—42no—
311C2#1963012yes—
312C2#1973211yescharacteristic
313C2#1975111yescharacteristic
314C2#2830111yescharacteristic
315C2#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. #10675C2semidirect
6. #10309C2semidirect
7. #9831C2semidirect
8. #9823C2semidirect

Maximal subgroups

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

subgroupindex
#331762
#278772
#258132
#212022
#106752
#103092
#98312
#98232
#81462