Finite Group Databasebeta

← #480715

Subgroups — #480715

Order 512.

990 conjugacy classes of subgroups; only the 202 normal subgroups are shown.

normal subgroups only

This group has 990 conjugacy classes of subgroups — too many to draw. Only its 202 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#35316C211yesmaximal, characteristic
2#17363C212yesmaximal
3#11558C218yesmaximal, complemented
4#11504C214yesmaximal, complemented
5C2×#536C412yes—
6C22×C4⋊Q8C411yescharacteristic
7C2×C4×#41C2211yescharacteristic
8C2×#432—22no—
9#2857—22no—
10#2621—22no—
11#2547—22no—
12#2546—22no—
13#2486C414yes—
14#2486C2214yes—
15#2310C2212yes—
16#2285C411yescharacteristic
17#1467—22no—
18#1466C2214yes—
19#1464—22no—
20#1464C2214yes—
21#1459—22no—
22#1459C2214yes—
23#1317C22116yes—
24C22×#41C2×C411yescharacteristic
25C23×Q8C2×C412yes—
26#536—48no—
27C2×C4⋊Q8—22no—
28C2×C4⋊Q8—21no—
29C2×C4⋊Q8—22no—
30C2×C4⋊Q8—21no—
31C2×C4⋊Q8—41no—
32C2×C4⋊Q8—42no—
33C2×C4⋊Q8—41no—
34C2×#124—22no—
35C2×#123C2×C418yes—
36C2×C22⋊Q8—24no—
37C4×#41—42no—
38C4×#41—41no—
39C4×#41—41no—
40C2×C4×Q8D412yes—
41C2×C4×D4C2×C412yes—
42C2×C4×D4D412yes—
43C2×C4×D4C2312yes—
44C2×#117C2×C412yes—
45C2×#117D412yes—
46C2×#117C2311yescharacteristic
47C22×C4⋊C4C2×C414yes—
48C22×C42C2×C411yescharacteristic
49C22×C42D412yes—
50#432—24no—
51#432—24no—
52#432—42no—
53#432—42no—
54C2×C4⋊C8—42no—
55C2×C22⋊C8—24no—
56C2×#97—42no—
57#398—24no—
58#398—24no—
59#397—24no—
60#397—24no—
61#391—24no—
62#391—24no—
63#389—24no—
64#389—24no—
65#384—21no—
66#382—21no—
67#381C2×C414yes—
68#381C2314yes—
69#380—24no—
70#380—24no—
71#380C2×C414yes—
72C4×C4⋊C4—22no—
73#324—24no—
74#324—24no—
75#323—24no—
76#323C2318yes—
77C2×#41—24no—
78C2×#41—41no—
79C2×#41—41no—
80C2×#41—42no—
81C22×Q8—22no—
82C22×Q8—22no—
83C22×Q8—44no—
84C22×Q8—42no—
85C22×Q8—42no—
86C22×Q8—44no—
87C22×Q8C22⋊C411yescharacteristic
88C22×D4C22⋊C411yescharacteristic
89C22×D4C22×C412yes—
90C23×C4C22⋊C412yes—
91C23×C4C22⋊C412yes—
92C23×C4C22×C411yescharacteristic
93C23×C4C22×C414yes—
94C23×C4C22×C412yes—
95C23×C4C2×D412yes—
96C22×C8—44no—
97C4⋊Q8—42no—
98C4⋊Q8—42no—
99C4⋊Q8—44no—
100C4⋊Q8—44no—
101C4⋊Q8—48no—
102C4⋊Q8—41no—
103C4⋊Q8—42no—
104C4⋊Q8—41no—
105#124—48no—
106#123—416no—
107C22⋊Q8—416no—
108C4×Q8—42no—
109C4×Q8—42no—
110C4×D4—42no—
111C4×D4—42no—
112C4×D4—42no—
113C4×D4—42no—
114C4×D4—42no—
115C4×D4—42no—
116#117—42no—
117#117—42no—
118#117—41no—
119#117—41no—
120#117—44no—
121C2×C4⋊C4—24no—
122C2×C4⋊C4—24no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—28no—
125C2×C4⋊C4—24no—
126C2×C4⋊C4—44no—
127C2×C4⋊C4—44no—
128C2×C4⋊C4C22⋊C412yes—
129C2×C4⋊C4C2×D412yes—
130C2×C4⋊C4C2×D412yes—
131C2×C22⋊C4—28no—
132C2×C22⋊C4C22⋊C412yes—
133C2×C22⋊C4C22×C414yes—
134C2×C22⋊C4C22×C411yescharacteristic
135C2×C22⋊C4C2×D412yes—
136C2×C22⋊C4C2411yescharacteristic
137C2×C42—21no—
138C2×C42—22no—
139C2×C42—21no—
140C2×C42—22no—
141C2×C42—22no—
142C2×C42—22no—
143C2×C42—22no—
144C2×C42—22no—
145C2×C42C22⋊C411yescharacteristic
146C2×C42C22⋊C412yes—
147C2×C42C22⋊C412yes—
148C2×C42C22⋊C411yescharacteristic
149C2×C42C2×D412yes—
150C2×C42C2×D412yes—
151C4⋊C8—44no—
152C4⋊C8—44no—
153C22⋊C8—28no—
154C22⋊C8—44no—
155#97—44no—
156#97—44no—
157#95—24no—
158#95—28no—
159#95—28no—
160C24C2×C22⋊C412yes—
161C24C23×C411yescharacteristic
162#41—48no—
163#41—48no—
164C2×Q8—22no—
165C2×Q8—41no—
166C2×Q8—41no—
167C2×Q8—44no—
168C2×Q8—44no—
169C2×Q8—44no—
170C2×Q8—44no—
171C2×Q8—42no—
172C2×Q8—42no—
173C2×Q8—44no—
174C2×Q8—88no—
175C2×Q8—88no—
176C2×D4—24no—
177C2×D4—24no—
178C2×D4—41no—
179C2×D4—42no—
180C2×D4—41no—
181C2×D4#10014yes—
182C22×C4—22no—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—22no—
186C22×C4—24no—
187C22×C4—22no—
188C22×C4—22no—
189C22×C4—22no—
190C22×C4—22no—
191C22×C4—22no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—24no—
196C22×C4—24no—
197C22×C4—28no—
198C22×C4—42no—
199C22×C4—44no—
200C22×C4—44no—
201C22×C4—44no—
202C22×C4—44no—
203C22×C4—44no—
204C22×C4C2×C22⋊C412yes—
205C22×C4C2×C22⋊C412yes—
206C22×C4C2×C22⋊C411yescharacteristic
207C22×C4C2×C22⋊C411yescharacteristic
208C22×C4C2×C22⋊C412yes—
209C22×C4C2×C22⋊C412yes—
210C22×C4C22×D412yes—
211C2×C8—48no—
212C2×C8—88no—
213C4⋊C4—42no—
214C4⋊C4—42no—
215C4⋊C4—42no—
216C4⋊C4—48no—
217C4⋊C4—48no—
218C4⋊C4—44no—
219C4⋊C4—416no—
220C4⋊C4—44no—
221C4⋊C4—44no—
222C22⋊C4—42no—
223C22⋊C4—41no—
224C22⋊C4—42no—
225C22⋊C4—41no—
226C22⋊C4—416no—
227C22⋊C4—44no—
228C42—22no—
229C42—22no—
230C42—22no—
231C42—22no—
232C42—21no—
233C42—21no—
234C42—22no—
235C42—22no—
236C42—22no—
237C42—22no—
238C42—44no—
239C42#9911yescharacteristic
240C42#9912yes—
241C42#9911yescharacteristic
242C23—21no—
243C23—22no—
244C23—21no—
245C23—22no—
246C23—42no—
247C23—42no—
248C23—42no—
249C23C2×#10014yes—
250C23C22×C22⋊C411yescharacteristic
251C23#56812yes—
252Q8—44no—
253Q8—84no—
254Q8—84no—
255Q8—88no—
256D4—44no—
257D4—48no—
258C2×C4—24no—
259C2×C4—24no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—24no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—44no—
272C2×C4—44no—
273C2×C4—44no—
274C2×C4—44no—
275C2×C4—44no—
276C2×C4—44no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—48no—
281C2×C4—44no—
282C2×C4—44no—
283C2×C4—44no—
284C2×C4—44no—
285C2×C4—82no—
286C2×C4—82no—
287C2×C4—88no—
288C2×C4C2×#9912yes—
289C2×C4C2×#9911yescharacteristic
290C2×C4C2×#9911yescharacteristic
291C2×C4C2×#9912yes—
292C2×C4C2×#10012yes—
293C8—88no—
294C22—22no—
295C22—22no—
296C22—24no—
297C22—42no—
298C22—44no—
299C22—42no—
300C22—42no—
301C22—42no—
302C22—42no—
303C22—82no—
304C22#287911yescharacteristic
305C22C22×#9911yescharacteristic
306C22C22×#10011yescharacteristic
307C4—22no—
308C4—21no—
309C4—21no—
310C4—22no—
311C4—42no—
312C4—42no—
313C4—42no—
314C4—42no—
315C4—84no—
316C4—84no—
317C4#212314yes—
318C2—21no—
319C2—21no—
320C2—42no—
321C2—44no—
322C2#1739611yescharacteristic
323C2C2×#212312yes—
subgroupG/H
center, socleC22C22×#99
commutatorC24C23×C4
Frattini, cosocleC2×C22⋊C4C24
prime radicalC22×#41C2×C4
LCSG ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C4×#41 ◁ G
LECSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #11558C2semidirect
4. #11504C2semidirect

Maximal subgroups

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

subgroupindex
#353162
#173632
#115582
#115042