Finite Group Databasebeta

← #392679

Subgroups — #392679

Order 512.

4622 conjugacy classes of subgroups; only the 454 normal subgroups are shown.

normal subgroups only

This group has 4622 conjugacy classes of subgroups — too many to draw. Only its 454 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
1C24×M16C211yesmaximal, characteristic, complemented
2C2×#2847C211yesmaximal, characteristic
3C2×#1900C216yesmaximal, complemented
4#20579C211yesmaximal, characteristic, complemented
5#10739C216yesmaximal, complemented
6C25×C4C2211yescharacteristic
7C23×M16—23no—
8C23×M16—224no—
9C23×M16C2216yes—
10C24×C8—21no—
11C22×#121C2213yes—
12C22×#120C412yes—
13C22×#100C2214yescomplemented
14C2×#407C2216yes—
15#2847—21no—
16#2847—23no—
17#2847—23no—
18#2847—21no—
19C2×#386C416yes—
20C2×#380C2213yes—
21#1900—224no—
22#1779—21no—
23#1779—23no—
24C2×#343C22112yescomplemented
25C26C2×C411yescharacteristic
26C23×D4C2×C416yes—
27C23×D4C2311yescharacteristic
28C24×C4—212no—
29C24×C4—212no—
30C24×C4C2×C413yes—
31C24×C4D411yescharacteristic
32C24×C4D413yes—
33C24×C4D411yescharacteristic
34C24×C4D413yes—
35C24×C4C2313yes—
36C22×M16—248no—
37C22×M16—2128no—
38C22×M16—224no—
39C22×M16—212no—
40C22×M16—26no—
41C22×M16—248no—
42C22×M16D4124yescomplemented
43C22×M16C2314yes—
44C22×M16C2314yes—
45C23×C8—224no—
46C23×C8—26no—
47C2×#121—23no—
48C2×#121—23no—
49C2×#121—23no—
50C2×#121—23no—
51C2×#121—212no—
52C2×#121—212no—
53C2×#120—26no—
54C2×#120—22no—
55C2×#120—48no—
56C22×C4⋊C4C2313yes—
57C22×C22⋊C4C2×C416yes—
58C22×C22⋊C4C2×C416yes—
59C2×#100—224no—
60#407—212no—
61#407—212no—
62C2×C22⋊C8—26no—
63C2×C22⋊C8—26no—
64#386—212no—
65#386—26no—
66#386—26no—
67#380—23no—
68#380—23no—
69#380—23no—
70#380—23no—
71C2×#95C2×C416yes—
72#343—224no—
73C25—212no—
74C25—24no—
75C25—46no—
76C25C22⋊C411yescharacteristic
77C25C22⋊C413yes—
78C25C22×C413yes—
79C25C22×C412yes—
80C22×D4—26no—
81C22×D4—26no—
82C22×D4—224no—
83C22×D4—21no—
84C22×D4—21no—
85C22×D4—26no—
86C22×D4—26no—
87C22×D4—412no—
88C22×D4—412no—
89C23×C4—232no—
90C23×C4—296no—
91C23×C4—212no—
92C23×C4—224no—
93C23×C4—26no—
94C23×C4—212no—
95C23×C4—26no—
96C23×C4—212no—
97C23×C4—224no—
98C23×C4—212no—
99C23×C4—26no—
100C23×C4—212no—
101C23×C4—26no—
102C23×C4—212no—
103C23×C4—212no—
104C23×C4C22⋊C416yes—
105C23×C4C22⋊C413yes—
106C23×C4C22⋊C413yes—
107C23×C4C22×C413yes—
108C23×C4C22×C416yes—
109C23×C4C2×D413yes—
110C23×C4C2×D413yes—
111C23×C4C2×D413yes—
112C23×C4C2×D413yes—
113C23×C4C2×D4112yes—
114C23×C4#41112yes—
115C23×C4C2411yescharacteristic
116C2×M16—2192no—
117C2×M16—296no—
118C2×M16—248no—
119C2×M16—224no—
120C2×M16—212no—
121C2×M16—296no—
122C2×M16—212no—
123C2×M16—212no—
124C2×M16C2×D4124yes—
125C22×C8—264no—
126C22×C8—248no—
127C22×C8—224no—
128C22×C8—24no—
129C22×C8—24no—
130#121—46no—
131#121—46no—
132#121—412no—
133#121—412no—
134#121—46no—
135#121—46no—
136#120—424no—
137#120—48no—
138C2×C4⋊C4—26no—
139C2×C4⋊C4—23no—
140C2×C4⋊C4—23no—
141C2×C4⋊C4—26no—
142C2×C22⋊C4—26no—
143C2×C22⋊C4—26no—
144C2×C22⋊C4—26no—
145C2×C22⋊C4—26no—
146C2×C22⋊C4—224no—
147C2×C22⋊C4—412no—
148#100—416no—
149C22⋊C8—412no—
150C22⋊C8—412no—
151#95—26no—
152#95—26no—
153C24—26no—
154C24—26no—
155C24—232no—
156C24—212no—
157C24—212no—
158C24—26no—
159C24—26no—
160C24—26no—
161C24—26no—
162C24—26no—
163C24—22no—
164C24—212no—
165C24—46no—
166C24—46no—
167C24—448no—
168C24—448no—
169C24—42no—
170C24—46no—
171C24#10014yescomplemented
172C24C2×C22⋊C413yes—
173C24C2×C22⋊C413yes—
174C24C4×D4112yes—
175C24C23×C411yescharacteristic
176C2×D4—412no—
177C2×D4—412no—
178C2×D4—412no—
179C2×D4—412no—
180C2×D4—412no—
181C2×D4—412no—
182C2×D4—44no—
183C2×D4—43no—
184C2×D4—46no—
185C2×D4—43no—
186C2×D4—412no—
187C2×D4—848no—
188C22×C4—28no—
189C22×C4—224no—
190C22×C4—296no—
191C22×C4—232no—
192C22×C4—296no—
193C22×C4—224no—
194C22×C4—224no—
195C22×C4—248no—
196C22×C4—212no—
197C22×C4—212no—
198C22×C4—26no—
199C22×C4—26no—
200C22×C4—26no—
201C22×C4—26no—
202C22×C4—224no—
203C22×C4—212no—
204C22×C4—212no—
205C22×C4—26no—
206C22×C4—26no—
207C22×C4—224no—
208C22×C4—248no—
209C22×C4—212no—
210C22×C4—212no—
211C22×C4—212no—
212C22×C4—212no—
213C22×C4—26no—
214C22×C4—212no—
215C22×C4—46no—
216C22×C4—46no—
217C22×C4C2×C22⋊C413yes—
218C22×C4C2×C22⋊C413yes—
219C22×C4C4×D4112yes—
220C22×C4#12016yes—
221C22×C4#12016yes—
222C22×C4#121112yes—
223C22×C4#121112yes—
224C22×C4#12316yes—
225C22×C4#12316yes—
226C22×C4C22×D416yes—
227C22×C4C22×D411yescharacteristic
228C22×C4C22×D411yescharacteristic
229C22×C4C2×#4116yes—
230M16—296no—
231M16—224no—
232M16—44no—
233M16—412no—
234M16#120116yescomplemented
235C2×C8—296no—
236C2×C8—224no—
237C2×C8—412no—
238C4⋊C4—43no—
239C4⋊C4—46no—
240C4⋊C4—43no—
241C22⋊C4—412no—
242C22⋊C4—412no—
243C22⋊C4—412no—
244C22⋊C4—412no—
245C23—28no—
246C23—224no—
247C23—21no—
248C23—23no—
249C23—26no—
250C23—224no—
251C23—26no—
252C23—224no—
253C23—23no—
254C23—23no—
255C23—26no—
256C23—224no—
257C23—26no—
258C23—224no—
259C23—412no—
260C23—424no—
261C23—424no—
262C23—424no—
263C23—424no—
264C23—4128no—
265C23—412no—
266C23—412no—
267C23—42no—
268C23—46no—
269C23—46no—
270C23—48no—
271C23—812no—
272C23—812no—
273C23#38616yes—
274C23#38616yes—
275C23C2×#10016yes—
276C23#41316yes—
277C23C22×C22⋊C411yescharacteristic
278C23C2×C4×D416yes—
279D4—824no—
280D4—824no—
281D4—81no—
282D4—84no—
283D4—83no—
284C2×C4—224no—
285C2×C4—224no—
286C2×C4—248no—
287C2×C4—26no—
288C2×C4—26no—
289C2×C4—224no—
290C2×C4—26no—
291C2×C4—26no—
292C2×C4—224no—
293C2×C4—224no—
294C2×C4—26no—
295C2×C4—26no—
296C2×C4—212no—
297C2×C4—212no—
298C2×C4—46no—
299C2×C4—46no—
300C2×C4—46no—
301C2×C4—812no—
302C2×C4#38616yes—
303C2×C4#38616yes—
304C2×C4C2×#12014yes—
305C2×C4C2×#12013yes—
306C2×C4C2×#12116yes—
307C2×C4C2×#12313yes—
308C2×C4#525112yes—
309C8—216no—
310C8—84no—
311C22—232no—
312C22—26no—
313C22—26no—
314C22—212no—
315C22—26no—
316C22—212no—
317C22—26no—
318C22—26no—
319C22—26no—
320C22—26no—
321C22—448no—
322C22—448no—
323C22—46no—
324C22—46no—
325C22—82no—
326C22—86no—
327C22—86no—
328C22—168no—
329C22#1900112yescomplemented
330C22C2×#38613yes—
331C22C4×#12014yes—
332C22C22×#10011yescharacteristic
333C22C2×#41313yes—
334C4—26no—
335C4—26no—
336C4—86no—
337C4#284611yescharacteristic
338C4#284714yes—
339C4#284913yes—
340C2—212no—
341C2—24no—
342C2—46no—
343C2—162no—
344C2#1469411yescharacteristic
345C2C2×#190013yes—
346C2#2098913yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#2847C2
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C25×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C24×M16C2semidirect
3. C2×#1900C2semidirect
4. #20579C2semidirect
5. #10739C2semidirect
13. C22×#100C22semidirect
24. C2×#343C22semidirect
42. C22×M16D4semidirect
171. C24#100semidirect
234. M16#120semidirect
329. C22#1900semidirect

Maximal subgroups

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

subgroupindex
C24×M162
C2×#28472
C2×#19002
#205792
#107392