Finite Group Databasebeta

← #375498

Subgroups — #375498

Order 512.

1530 conjugacy classes of subgroups.

The subgroup lattice has 349 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2627C211yesmaximal, characteristic
2#13795C211yesmaximal, characteristic
3#13776C211yesmaximal, characteristic
4#11381C212yesmaximal
5#11353C212yesmaximal
6#10912C214yesmaximal
7#10736C214yesmaximal
8C22×C22⋊Q8C2211yescharacteristic
9C22×#117C2212yes—
10C23×C4⋊C4C2211yescharacteristic
11#2627—24no—
12#2627—24no—
13C2×#400C412yescomplemented
14C2×#398C414yescomplemented
15C2×#395C412yescomplemented
16C2×#385C2212yes—
17C2×#381C2212yes—
18C2×#381C2212yes—
19C2×#380C2211yescharacteristic
20C2×C4×C22⋊C4C2212yes—
21C2×#342C2218yes—
22C2×#342C2218yes—
23#1436—24no—
24C4⋊#95—24no—
25C22⋊#95C2212yes—
26C22⋊#95C2212yes—
27C22⋊#95C2212yes—
28C23×Q8C2×C411yescharacteristic
29C24×C4C2311yescharacteristic
30C24×C4C2311yescharacteristic
31C24×C4C2311yescharacteristic
32C2×C22⋊Q8—22no—
33C2×C22⋊Q8—22no—
34C2×C22⋊Q8—44no—
35C2×#117—28no—
36C2×#117—28no—
37C2×#117—44no—
38C22×C4⋊C4—22no—
39C22×C4⋊C4—44no—
40C22×C4⋊C4C2×C412yes—
41C22×C4⋊C4C2×C411yescharacteristic
42C22×C4⋊C4C2×C414yes—
43C22×C4⋊C4C2×C412yes—
44C22×C4⋊C4C2×C412yes—
45C22×C4⋊C4C2×C412yes—
46C22×C4⋊C4D414yes—
47C22×C4⋊C4D414yes—
48C22×C22⋊C4C2312yes—
49C22×C22⋊C4C2312yes—
50C22×C22⋊C4C2314yes—
51C22×C22⋊C4C2314yes—
52C22×C42—22no—
53C22×C42—22no—
54#400—28no—
55#398—216no—
56#395—28no—
57#385—24no—
58#385—24no—
59#381—24no—
60#381—24no—
61#381—24no—
62#381Q818yescomplemented
63#380—22no—
64#380—22no—
65C4×C22⋊C4—44no—
66C2×#95—24no—
67C2×#95—22no—
68C2×#95—24no—
69C2×#95—24no—
70C2×#95—24no—
71C2×#95C2×C414yes—
72C2×#95C2×C412yes—
73C2×#95C2×C414yes—
74C2×#95C2×C414yes—
75#342—216no—
76#342—48no—
77C25C2411yescharacteristic
78C22×Q8—22no—
79C22×Q8—42no—
80C22×Q8—82no—
81C23×C4—24no—
82C23×C4—24no—
83C23×C4—24no—
84C23×C4—24no—
85C23×C4—24no—
86C23×C4—24no—
87C23×C4—82no—
88C23×C4—82no—
89C23×C4C22×C412yes—
90C23×C4C22×C412yes—
91C23×C4C22×C414yes—
92C23×C4C22×C412yes—
93C23×C4C22×C411yescharacteristic
94C23×C4C22×C412yes—
95C23×C4C22×C411yescharacteristic
96C23×C4C2×D412yes—
97C23×C4C2×D412yes—
98C23×C4C2×D412yes—
99C23×C4C2×D412yes—
100C23×C4C2×D412yes—
101C23×C4C2×D412yes—
102C23×C4C2×Q814yes—
103C23×C4#4114yes—
104C22⋊Q8—48no—
105C22⋊Q8—48no—
106#117—432no—
107C2×C4⋊C4—28no—
108C2×C4⋊C4—24no—
109C2×C4⋊C4—22no—
110C2×C4⋊C4—24no—
111C2×C4⋊C4—28no—
112C2×C4⋊C4—44no—
113C2×C4⋊C4—44no—
114C2×C4⋊C4—44no—
115C2×C4⋊C4—44no—
116C2×C4⋊C4—44no—
117C2×C4⋊C4—44no—
118C2×C4⋊C4—44no—
119C2×C4⋊C4—42no—
120C2×C4⋊C4—44no—
121C2×C4⋊C4—44no—
122C2×C4⋊C4—82no—
123C2×C4⋊C4—84no—
124C2×C4⋊C4C22⋊C418yes—
125C2×C4⋊C4C22⋊C418yes—
126C2×C22⋊C4—28no—
127C2×C22⋊C4—28no—
128C2×C22⋊C4—28no—
129C2×C22⋊C4—28no—
130C2×C22⋊C4—22no—
131C2×C22⋊C4—22no—
132C2×C22⋊C4—22no—
133C2×C22⋊C4—22no—
134C2×C22⋊C4—24no—
135C2×C22⋊C4—28no—
136C2×C22⋊C4—24no—
137C2×C22⋊C4—28no—
138C2×C22⋊C4C2×Q818yes—
139C2×C22⋊C4#4118yes—
140C2×C42—28no—
141C2×C42—44no—
142C2×C42—44no—
143C2×C42—44no—
144C2×C42—84no—
145#95—24no—
146#95—28no—
147#95—28no—
148#95—28no—
149#95—216no—
150#95—44no—
151#95—84no—
152#95—84no—
153#95—84no—
154C24—42no—
155C24—42no—
156C24C23×C411yescharacteristic
157C24C22×D411yescharacteristic
158C24C22×D411yescharacteristic
159C24C22×Q812yes—
160C24C2×#4112yes—
161C24C2×#4114yes—
162C24H(5,𝔽2)14yes—
163C2×Q8—44no—
164C2×Q8—84no—
165C2×Q8—88no—
166C22×C4—24no—
167C22×C4—24no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—24no—
171C22×C4—22no—
172C22×C4—22no—
173C22×C4—24no—
174C22×C4—216no—
175C22×C4—216no—
176C22×C4—22no—
177C22×C4—22no—
178C22×C4—22no—
179C22×C4—22no—
180C22×C4—24no—
181C22×C4—24no—
182C22×C4—24no—
183C22×C4—24no—
184C22×C4—22no—
185C22×C4—24no—
186C22×C4—22no—
187C22×C4—28no—
188C22×C4—28no—
189C22×C4—28no—
190C22×C4—24no—
191C22×C4—24no—
192C22×C4—44no—
193C22×C4—42no—
194C22×C4—42no—
195C22×C4—44no—
196C22×C4—44no—
197C22×C4—44no—
198C22×C4—44no—
199C22×C4—48no—
200C22×C4—48no—
201C22×C4—48no—
202C22×C4—48no—
203C22×C4—44no—
204C22×C4—44no—
205C22×C4—44no—
206C22×C4—44no—
207C22×C4—44no—
208C22×C4—44no—
209C22×C4—44no—
210C22×C4—44no—
211C22×C4—88no—
212C22×C4—88no—
213C22×C4C2×C22⋊C414yes—
214C22×C4C2×C22⋊C412yes—
215C22×C4C2×C22⋊C412yes—
216C22×C4C2×C22⋊C414yes—
217C22×C4C4×Q818yes—
218C22×C4C22⋊Q818yes—
219C22×C4#12414yes—
220C22×C4C4⋊Q814yes—
221C4⋊C4—48no—
222C4⋊C4—48no—
223C4⋊C4—416no—
224C4⋊C4—48no—
225C4⋊C4—44no—
226C4⋊C4—84no—
227C4⋊C4—88no—
228C22⋊C4—216no—
229C22⋊C4—216no—
230C22⋊C4—44no—
231C22⋊C4—44no—
232C22⋊C4—44no—
233C22⋊C4—44no—
234C22⋊C4—48no—
235C22⋊C4—48no—
236C42—416no—
237C42—88no—
238C23—22no—
239C23—22no—
240C23—22no—
241C23—22no—
242C23—22no—
243C23—22no—
244C23—44no—
245C23—44no—
246C23—44no—
247C23—48no—
248C23—44no—
249C23C22×C22⋊C411yescharacteristic
250C23C2×C4×Q812yes—
251C23C4×#4112yes—
252C23#51812yes—
253C23C2×C22⋊Q812yes—
254C23#52512yes—
255C23C2×#12411yescharacteristic
256C23C2×C4⋊Q811yescharacteristic
257C23#53212yes—
258C23#53412yes—
259C23#53412yes—
260C23#55118yes—
261C23#55418yes—
262Q8—88no—
263C2×C4—24no—
264C2×C4—24no—
265C2×C4—24no—
266C2×C4—216no—
267C2×C4—24no—
268C2×C4—24no—
269C2×C4—216no—
270C2×C4—24no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—24no—
274C2×C4—28no—
275C2×C4—24no—
276C2×C4—24no—
277C2×C4—44no—
278C2×C4—44no—
279C2×C4—44no—
280C2×C4—44no—
281C2×C4—416no—
282C2×C4—416no—
283C2×C4—48no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—48no—
287C2×C4—44no—
288C2×C4—44no—
289C2×C4—44no—
290C2×C4—44no—
291C2×C4—44no—
292C2×C4—84no—
293C2×C4—84no—
294C2×C4—84no—
295C2×C4—84no—
296C2×C4—84no—
297C2×C4—88no—
298C2×C4—88no—
299C2×C4—84no—
300C2×C4—84no—
301C2×C4—84no—
302C2×C4—88no—
303C2×C4—84no—
304C2×C4—84no—
305C2×C4#39118yes—
306C22—24no—
307C22—22no—
308C22—24no—
309C22—22no—
310C22—24no—
311C22—24no—
312C22—24no—
313C22—22no—
314C22—22no—
315C22—22no—
316C22—22no—
317C22—22no—
318C22—22no—
319C22—48no—
320C22—48no—
321C22C2×#39111yescharacteristic
322C22#229711yescharacteristic
323C22#230811yescharacteristic
324C22#236814yes—
325C22#256112yes—
326C22#256414yes—
327C22#258312yes—
328C22#288312yes—
329C22#288412yes—
330C4—28no—
331C4—42no—
332C4—44no—
333C4—42no—
334C4—88no—
335C4—88no—
336C4—84no—
337C4—82no—
338C4—82no—
339C2—22no—
340C2—24no—
341C2—22no—
342C2—24no—
343C2#1415911yescharacteristic
344C2#2133412yes—
345C2#2142312yes—
346C2#2174911yescharacteristic
347C2#2175111yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, prime radical, cosocleC25C24
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4⋊C4 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
13. C2×#400C4semidirect
14. C2×#398C4semidirect
15. C2×#395C4semidirect
62. #381Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#26272
#137952
#137762
#113812
#113532
#109122
#107362