Finite Group Databasebeta

← #496313

Subgroups — #496313

Order 512.

1090 conjugacy classes of subgroups; only the 202 normal subgroups are shown.

normal subgroups only

This group has 1090 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#20723C212yesmaximal
3#13224C218yesmaximal, complemented
4#12698C214yesmaximal
5C2×#536C2212yes—
6C22×C4⋊Q8C411yescharacteristic, complemented
7C2×C4×#41C2211yescharacteristic
8#2857—22no—
9#2621—22no—
10#2547C2214yes—
11#2546C414yescomplemented
12#2486—22no—
13#2486—22no—
14#2310C412yescomplemented
15#2285C411yescharacteristic, complemented
16#2283—22no—
17C2×#354C22116yes—
18#1820—22no—
19#1799—22no—
20#1799C2214yes—
21#1782C2214yes—
22C4×#99—22no—
23C4×#99C2214yes—
24C22×#41C2311yescharacteristic
25C23×Q8C2×C412yes—
26#536—48no—
27C2×C4⋊Q8—21no—
28C2×C4⋊Q8—21no—
29C2×C4⋊Q8—41no—
30C2×C4⋊Q8—42no—
31C2×C4⋊Q8—41no—
32C2×C4⋊Q8C2×C414yes—
33C2×C4⋊Q8C2×C414yes—
34C2×#124C2314yes—
35C2×#123—24no—
36C2×C22⋊Q8—24no—
37C4×#41—42no—
38C4×#41—41no—
39C4×#41—41no—
40C2×C4×Q8C2×C412yes—
41C2×C4×D4D412yes—
42C2×C4×D4D412yes—
43C2×C4×D4C2312yes—
44C2×#117C2×C411yescharacteristic
45C2×#117C2×C412yes—
46C2×#117D412yes—
47C22×C4⋊C4—22no—
48C22×C42C2×C411yescharacteristic
49C22×C42D412yes—
50C2×#99—24no—
51C2×#99C2318yes—
52#398—24no—
53#398—24no—
54#397—24no—
55#397—24no—
56#391—24no—
57#391C2×C418yes—
58#389—24no—
59#389—24no—
60#384—21no—
61#383—42no—
62#382—21no—
63#382—42no—
64#381—22no—
65#381—22no—
66#381—24no—
67#380—22no—
68#380—24no—
69#380—24no—
70#380—24no—
71C4×C4⋊C4—42no—
72C4×C4⋊C4C2×C414yes—
73C4×C22⋊C4—24no—
74#376—42no—
75#354—416no—
76#342—24no—
77#342—24no—
78C2×#41—24no—
79C2×#41—41no—
80C2×#41—41no—
81C2×#41—42no—
82C22×Q8—22no—
83C22×Q8—44no—
84C22×Q8—42no—
85C22×Q8—42no—
86C22×Q8—44no—
87C22×Q8C22×C414yes—
88C22×Q8C22×C411yescharacteristic
89C22×D4C2×D412yes—
90C22×D4C2411yescharacteristic
91C23×C4—22no—
92C23×C4C22⋊C412yes—
93C23×C4C22×C411yescharacteristic
94C23×C4C2×D412yes—
95C23×C4C2×D412yes—
96C23×C4C2×D412yes—
97C4⋊Q8—48no—
98C4⋊Q8—42no—
99C4⋊Q8—42no—
100C4⋊Q8—44no—
101C4⋊Q8—44no—
102C4⋊Q8—41no—
103C4⋊Q8—42no—
104C4⋊Q8—41no—
105#124—216no—
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—41no—
117#117—41no—
118#117—44no—
119#117—42no—
120#117—42no—
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⋊C4—44no—
129C2×C4⋊C4C22⋊C412yes—
130C2×C4⋊C4C22⋊C412yes—
131C2×C4⋊C4C22×C412yes—
132C2×C22⋊C4—22no—
133C2×C22⋊C4—28no—
134C2×C22⋊C4—28no—
135C2×C22⋊C4C22⋊C411yescharacteristic
136C2×C22⋊C4C22⋊C411yescharacteristic
137C2×C22⋊C4C2×D412yes—
138C2×C22⋊C4C2×D412yes—
139C2×C42—21no—
140C2×C42—22no—
141C2×C42—21no—
142C2×C42—22no—
143C2×C42—22no—
144C2×C42—22no—
145C2×C42—22no—
146C2×C42—44no—
147C2×C42C22⋊C412yes—
148C2×C42C22⋊C411yescharacteristic
149C2×C42C22⋊C412yes—
150C2×C42C22⋊C412yes—
151C2×C42C22⋊C411yescharacteristic
152C2×C42C22×C414yes—
153C2×C42C22×C412yes—
154#99—216no—
155#99—48no—
156#95—24no—
157#95—28no—
158#95—28no—
159#95—44no—
160#95—44no—
161#95—44no—
162#95—44no—
163C24C2×C22⋊C411yescharacteristic
164C24C22×D412yes—
165#41—48no—
166#41—48no—
167C2×Q8—22no—
168C2×Q8—28no—
169C2×Q8—41no—
170C2×Q8—41no—
171C2×Q8—44no—
172C2×Q8—44no—
173C2×Q8—44no—
174C2×Q8—44no—
175C2×Q8—42no—
176C2×Q8—42no—
177C2×Q8—88no—
178C2×Q8—88no—
179C2×D4—24no—
180C2×D4—24no—
181C2×D4—42no—
182C2×D4—41no—
183C2×D4—41no—
184C2×D4H(5,𝔽2)14yes—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—21no—
188C22×C4—21no—
189C22×C4—24no—
190C22×C4—22no—
191C22×C4—22no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—22no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—24no—
199C22×C4—24no—
200C22×C4—42no—
201C22×C4—44no—
202C22×C4—44no—
203C22×C4—44no—
204C22×C4—44no—
205C22×C4—44no—
206C22×C4—44no—
207C22×C4—48no—
208C22×C4C2×C22⋊C412yes—
209C22×C4C2×C22⋊C411yescharacteristic
210C22×C4C2×C22⋊C412yes—
211C22×C4C2×C22⋊C412yes—
212C22×C4C2×C22⋊C412yes—
213C22×C4C2×C22⋊C412yes—
214C22×C4C23×C411yescharacteristic
215C4⋊C4—42no—
216C4⋊C4—42no—
217C4⋊C4—42no—
218C4⋊C4—48no—
219C4⋊C4—48no—
220C4⋊C4—44no—
221C4⋊C4—416no—
222C4⋊C4—44no—
223C4⋊C4—44no—
224C4⋊C4—88no—
225C22⋊C4—41no—
226C22⋊C4—41no—
227C22⋊C4—42no—
228C22⋊C4—42no—
229C22⋊C4—44no—
230C22⋊C4—416no—
231C22⋊C4—48no—
232C22⋊C4—48no—
233C42—22no—
234C42—22no—
235C42—21no—
236C42—21no—
237C42—22no—
238C42—22no—
239C42—22no—
240C42—22no—
241C42—22no—
242C42—22no—
243C42—28no—
244C42—88no—
245C42#9911yescharacteristic
246C42#9912yes—
247C42#9911yescharacteristic
248C23—21no—
249C23—21no—
250C23—22no—
251C23—22no—
252C23—42no—
253C23—42no—
254C23—42no—
255C23#41012yes—
256C23C22×C22⋊C411yescharacteristic
257C23#53514yes—
258Q8—44no—
259Q8—88no—
260Q8—84no—
261Q8—84no—
262D4—44no—
263D4—48no—
264C2×C4—24no—
265C2×C4—24no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—24no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—44no—
278C2×C4—44no—
279C2×C4—44no—
280C2×C4—44no—
281C2×C4—44no—
282C2×C4—44no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—48no—
287C2×C4—44no—
288C2×C4—44no—
289C2×C4—44no—
290C2×C4—44no—
291C2×C4—82no—
292C2×C4—82no—
293C2×C4—88no—
294C2×C4—88no—
295C2×C4—88no—
296C2×C4—88no—
297C2×C4C2×#9912yes—
298C2×C4C2×#9911yescharacteristic
299C2×C4C2×#9911yescharacteristic
300C2×C4C2×#9912yes—
301C2×C4#51812yes—
302C22—22no—
303C22—22no—
304C22—24no—
305C22—44no—
306C22—42no—
307C22—42no—
308C22—42no—
309C22—42no—
310C22—42no—
311C22—82no—
312C22#230911yescharacteristic
313C22C22×#9911yescharacteristic
314C22C2×#41011yescharacteristic
315C4—21no—
316C4—21no—
317C4—22no—
318C4—22no—
319C4—42no—
320C4—42no—
321C4—42no—
322C4—42no—
323C4—84no—
324C4—84no—
325C4—168no—
326C4#212914yes—
327C2—21no—
328C2—21no—
329C2—44no—
330C2—42no—
331C2#2179311yescharacteristic
332C2C2×#212912yes—
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC22×D4C24
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #13224C2semidirect
6. C22×C4⋊Q8C4semidirect
11. #2546C4semidirect
14. #2310C4semidirect
15. #2285C4semidirect

Maximal subgroups

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

subgroupindex
#353162
#207232
#132242
#126982