Finite Group Databasebeta

← #496401

Subgroups — #496401

Order 512.

938 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35532C211yesmaximal, characteristic
2#20725C212yesmaximal
3#13300C214yesmaximal, complemented
4#13298C214yesmaximal, complemented
5#12859C212yesmaximal
6#12857C212yesmaximal
7C2×#544C411yescharacteristic, complemented
8C2×#520C2211yescharacteristic
9#2567C412yescomplemented
10#2566C2212yes—
11#2565C412yescomplemented
12#2542C2212yes—
13#2449—22no—
14#2449—22no—
15#2441—22no—
16#2440—22no—
17#2317C411yescharacteristic, complemented
18#2313—22no—
19#2311C2211yescharacteristic
20#2297C2211yescharacteristic
21#2296C411yescharacteristic, complemented
22C2×#384C411yescharacteristic, complemented
23#1820—22no—
24#1799—22no—
25#1799C2214yes—
26#1782C2214yes—
27C4×#99—22no—
28C4×#99C2214yes—
29#1393C2214yes—
30#1393C2214yes—
31#1393C2218yes—
32C22×#41C2311yescharacteristic
33#544—44no—
34C2×C4⋊Q8—21no—
35C2×C4⋊Q8C2×C412yes—
36C2×#125—21no—
37C2×#125C2×C412yes—
38C2×C22⋊Q8—22no—
39C2×C22⋊Q8—22no—
40C2×C22⋊Q8—22no—
41#520—44no—
42C2×C4×Q8C2×C412yes—
43C2×C4×D4D412yes—
44C2×C4×D4D412yes—
45C2×C4×D4C2312yes—
46C2×#117C2×C411yescharacteristic
47C2×#117C2×C411yescharacteristic
48C2×#117C2×C411yescharacteristic
49C2×#117D412yes—
50C22×C4⋊C4—21no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4C2×C411yescharacteristic
53C22×C4⋊C4C2×C411yescharacteristic
54C22×C4⋊C4D412yes—
55C22×C42C2×C411yescharacteristic
56C2×#99—24no—
57C2×#99C2318yes—
58#399—22no—
59#399—22no—
60#398—22no—
61#398—22no—
62#396—22no—
63#396—22no—
64#395—22no—
65#395—22no—
66#391C2×C411yescharacteristic
67#391C2×C412yes—
68#391C2×C411yescharacteristic
69#390C2311yescharacteristic
70#390C2312yes—
71#390C2311yescharacteristic
72#389—22no—
73#389—21no—
74#389—21no—
75#389—22no—
76#389—22no—
77#387—22no—
78#387—22no—
79#387—48no—
80#386—22no—
81#386—22no—
82#385—21no—
83#385—21no—
84#385—22no—
85#385—22no—
86#385—22no—
87#384—22no—
88#384C2×C412yes—
89#384C2×C412yes—
90#384C2×C412yes—
91#382C2×C412yes—
92#381—24no—
93#380—22no—
94#380—24no—
95C4×C4⋊C4C2×C412yes—
96C4×C4⋊C4C2×C412yes—
97C4×C22⋊C4—22no—
98C4×C22⋊C4—24no—
99#342—24no—
100#342—24no—
101C2×#41—24no—
102C2×#41—42no—
103C2×#41—42no—
104C22×Q8—22no—
105C22×Q8C22×C411yescharacteristic
106C22×D4C2×D412yes—
107C22×D4C2411yescharacteristic
108C23×C4—21no—
109C23×C4—21no—
110C23×C4C22⋊C411yescharacteristic
111C23×C4C22⋊C411yescharacteristic
112C23×C4C22×C411yescharacteristic
113C23×C4C2×D412yes—
114C23×C4C2×D412yes—
115C23×C4C2×D412yes—
116C4⋊Q8—44no—
117C4⋊Q8—44no—
118#125—44no—
119#125—44no—
120C22⋊Q8—48no—
121C22⋊Q8—84no—
122C22⋊Q8—84no—
123C4×Q8—44no—
124C4×D4—44no—
125C4×D4—44no—
126C4×D4—44no—
127#117—24no—
128#117—44no—
129#117—42no—
130#117—42no—
131C2×C4⋊C4—28no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—21no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—42no—
146C2×C4⋊C4—42no—
147C2×C4⋊C4—44no—
148C2×C4⋊C4—82no—
149C2×C4⋊C4—82no—
150C2×C4⋊C4C22⋊C412yes—
151C2×C4⋊C4C22⋊C412yes—
152C2×C4⋊C4C22⋊C412yes—
153C2×C4⋊C4C22⋊C411yescharacteristic
154C2×C4⋊C4C22⋊C411yescharacteristic
155C2×C4⋊C4C22×C412yes—
156C2×C4⋊C4C22×C411yescharacteristic
157C2×C4⋊C4C22×C411yescharacteristic
158C2×C4⋊C4C22×C412yes—
159C2×C22⋊C4—21no—
160C2×C22⋊C4—21no—
161C2×C22⋊C4—22no—
162C2×C22⋊C4—22no—
163C2×C22⋊C4—28no—
164C2×C22⋊C4—22no—
165C2×C22⋊C4—22no—
166C2×C22⋊C4C22⋊C412yes—
167C2×C22⋊C4C2×D412yes—
168C2×C22⋊C4C2×D412yes—
169C2×C42—21no—
170C2×C42—21no—
171C2×C42—42no—
172C2×C42—44no—
173C2×C42C22⋊C412yes—
174C2×C42C22⋊C412yes—
175C2×C42C22×C412yes—
176C2×C42C22×C412yes—
177C2×C42C22×C411yescharacteristic
178C2×C42C22×C411yescharacteristic
179#99—216no—
180#99—48no—
181#95—22no—
182#95—22no—
183#95—22no—
184#95—22no—
185#95—22no—
186#95—22no—
187#95—22no—
188#95—22no—
189#95—44no—
190#95—44no—
191#95—48no—
192C24C2×C22⋊C411yescharacteristic
193C24C22×D412yes—
194#41—416no—
195C2×Q8—22no—
196C2×Q8—41no—
197C2×Q8—41no—
198C2×Q8—44no—
199C2×Q8—82no—
200C2×Q8—82no—
201C2×D4—24no—
202C2×D4—24no—
203C2×D4—42no—
204C2×D4—41no—
205C2×D4—41no—
206C2×D42−1+414yes—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—24no—
213C22×C4—24no—
214C22×C4—24no—
215C22×C4—24no—
216C22×C4—21no—
217C22×C4—22no—
218C22×C4—21no—
219C22×C4—22no—
220C22×C4—42no—
221C22×C4—41no—
222C22×C4—41no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—48no—
228C22×C4—82no—
229C22×C4—82no—
230C22×C4—81no—
231C22×C4—81no—
232C22×C4—81no—
233C22×C4—81no—
234C22×C4C2×C22⋊C412yes—
235C22×C4C2×C22⋊C411yescharacteristic
236C22×C4C2×C22⋊C412yes—
237C22×C4C2×C22⋊C412yes—
238C22×C4C2×C22⋊C412yes—
239C22×C4C2×C22⋊C412yes—
240C22×C4C23×C411yescharacteristic
241C4⋊C4—24no—
242C4⋊C4—24no—
243C4⋊C4—24no—
244C4⋊C4—22no—
245C4⋊C4—42no—
246C4⋊C4—42no—
247C4⋊C4—41no—
248C4⋊C4—41no—
249C4⋊C4—44no—
250C4⋊C4—44no—
251C4⋊C4—44no—
252C4⋊C4—42no—
253C4⋊C4—44no—
254C4⋊C4—44no—
255C4⋊C4—44no—
256C4⋊C4—42no—
257C4⋊C4—44no—
258C4⋊C4—82no—
259C4⋊C4—82no—
260C4⋊C4—82no—
261C4⋊C4—82no—
262C4⋊C4—164no—
263C4⋊C4#9914yes—
264C22⋊C4—24no—
265C22⋊C4—42no—
266C22⋊C4—42no—
267C22⋊C4—44no—
268C22⋊C4—44no—
269C22⋊C4—42no—
270C22⋊C4—42no—
271C22⋊C4—48no—
272C22⋊C4—48no—
273C22⋊C4—82no—
274C22⋊C4—82no—
275C42—24no—
276C42—42no—
277C42—42no—
278C42—41no—
279C42—41no—
280C42—42no—
281C42—42no—
282C42—42no—
283C42—164no—
284C23—21no—
285C23—21no—
286C23—22no—
287C23—22no—
288C23—41no—
289C23—41no—
290C23—42no—
291C23—42no—
292C23#41012yes—
293C23C22×C22⋊C411yescharacteristic
294C23#53614yes—
295Q8—44no—
296Q8—164no—
297D4—44no—
298D4—48no—
299C2×C4—28no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—42no—
313C2×C4—48no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—48no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—84no—
333C2×C4—84no—
334C2×C4—88no—
335C2×C4—88no—
336C2×C4—88no—
337C2×C4—161no—
338C2×C4—161no—
339C2×C4—162no—
340C2×C4C2×#9914yes—
341C2×C4C2×#9912yes—
342C2×C4#51912yes—
343C22—21no—
344C22—21no—
345C22—22no—
346C22—24no—
347C22—42no—
348C22—44no—
349C22—42no—
350C22—42no—
351C22—41no—
352C22—42no—
353C22—41no—
354C22—82no—
355C22#231011yescharacteristic
356C22C22×#9911yescharacteristic
357C22C2×#41011yescharacteristic
358C4—22no—
359C4—22no—
360C4—24no—
361C4—42no—
362C4—42no—
363C4—42no—
364C4—42no—
365C4—42no—
366C4—42no—
367C4—82no—
368C4—82no—
369C4—81no—
370C4—81no—
371C4—168no—
372C2—21no—
373C2—21no—
374C2—44no—
375C2—42no—
376C2#2182511yescharacteristic
377C2#3335911yescharacteristic
378C2#3336011yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC22×D4C24
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C4⋊C4 ◁ C2×#520 ◁ G
LECSG ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #13300C2semidirect
4. #13298C2semidirect
7. C2×#544C4semidirect
9. #2567C4semidirect
11. #2565C4semidirect
17. #2317C4semidirect
21. #2296C4semidirect
22. C2×#384C4semidirect

Maximal subgroups

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

subgroupindex
#355322
#207252
#133002
#132982
#128592
#128572