Finite Group Databasebeta

← #496402

Subgroups — #496402

Order 512.

896 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35620C211yesmaximal, characteristic
2#20725C212yesmaximal
3#13300C214yesmaximal, complemented
4#13299C214yesmaximal, complemented
5#12859C212yesmaximal
6#12857C212yesmaximal
7C2×#544C411yescharacteristic, complemented
8C2×#520C2211yescharacteristic
9#2568C412yescomplemented
10#2566C2212yes—
11#2565C412yescomplemented
12#2563C2212yes—
13#2450—22no—
14#2450—22no—
15#2442—22no—
16#2439—22no—
17#2317C411yescharacteristic, complemented
18#2313—22no—
19#2311C2211yescharacteristic
20#2297C2211yescharacteristic
21#2296C411yescharacteristic, complemented
22C2×#382C411yescharacteristic, 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—
39#520—44no—
40C2×C4×Q8C2×C412yes—
41C2×C4×D4D412yes—
42C2×C4×D4D412yes—
43C2×C4×D4C2312yes—
44C2×#117C2×C411yescharacteristic
45C2×#117C2×C411yescharacteristic
46C2×#117C2×C411yescharacteristic
47C2×#117D412yes—
48C22×C4⋊C4C2×C411yescharacteristic
49C22×C4⋊C4C2×C411yescharacteristic
50C22×C4⋊C4D412yes—
51C22×C42C2×C411yescharacteristic
52C2×#99—24no—
53C2×#99C2318yes—
54#400—22no—
55#400—22no—
56#398—22no—
57#398—22no—
58#396—22no—
59#396—22no—
60#393—22no—
61#393—22no—
62#391—22no—
63#391—22no—
64#391C2×C411yescharacteristic
65#391C2×C412yes—
66#391C2×C411yescharacteristic
67#390C2311yescharacteristic
68#390C2312yes—
69#390C2311yescharacteristic
70#389—22no—
71#389—21no—
72#389—21no—
73#387—22no—
74#387—22no—
75#387—48no—
76#386—22no—
77#386—22no—
78#385—21no—
79#385—21no—
80#385—22no—
81#385—22no—
82#385—22no—
83#384C2×C412yes—
84#382—22no—
85#382C2×C412yes—
86#382C2×C412yes—
87#382C2×C412yes—
88#381—24no—
89#380—22no—
90#380—22no—
91#380—22no—
92#380—24no—
93C4×C4⋊C4C2×C412yes—
94C4×C4⋊C4C2×C412yes—
95C4×C22⋊C4—22no—
96C4×C22⋊C4—24no—
97C2×#95—21no—
98C2×#95—21no—
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—
121C4×Q8—44no—
122C4×D4—44no—
123C4×D4—44no—
124C4×D4—44no—
125#117—24no—
126#117—44no—
127#117—42no—
128#117—42no—
129C2×C4⋊C4—28no—
130C2×C4⋊C4—21no—
131C2×C4⋊C4—21no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—42no—
140C2×C4⋊C4—42no—
141C2×C4⋊C4—44no—
142C2×C4⋊C4C22⋊C412yes—
143C2×C4⋊C4C22⋊C412yes—
144C2×C4⋊C4C22⋊C412yes—
145C2×C4⋊C4C22⋊C411yescharacteristic
146C2×C4⋊C4C22⋊C411yescharacteristic
147C2×C4⋊C4C22×C412yes—
148C2×C4⋊C4C22×C411yescharacteristic
149C2×C4⋊C4C22×C411yescharacteristic
150C2×C4⋊C4C22×C412yes—
151C2×C22⋊C4—21no—
152C2×C22⋊C4—21no—
153C2×C22⋊C4—28no—
154C2×C22⋊C4—22no—
155C2×C22⋊C4—22no—
156C2×C22⋊C4—22no—
157C2×C22⋊C4—22no—
158C2×C22⋊C4C22⋊C412yes—
159C2×C22⋊C4C2×D412yes—
160C2×C22⋊C4C2×D412yes—
161C2×C42—21no—
162C2×C42—21no—
163C2×C42—42no—
164C2×C42—44no—
165C2×C42C22⋊C412yes—
166C2×C42C22⋊C412yes—
167C2×C42C22×C412yes—
168C2×C42C22×C412yes—
169C2×C42C22×C411yescharacteristic
170C2×C42C22×C411yescharacteristic
171#99—216no—
172#99—48no—
173#95—22no—
174#95—22no—
175#95—22no—
176#95—22no—
177#95—22no—
178#95—22no—
179#95—22no—
180#95—22no—
181#95—22no—
182#95—22no—
183#95—22no—
184#95—22no—
185#95—44no—
186#95—44no—
187#95—48no—
188C24C2×C22⋊C411yescharacteristic
189C24C22×D412yes—
190#41—416no—
191C2×Q8—22no—
192C2×Q8—41no—
193C2×Q8—41no—
194C2×Q8—44no—
195C2×Q8—82no—
196C2×Q8—82no—
197C2×D4—24no—
198C2×D4—24no—
199C2×D4—42no—
200C2×D4—41no—
201C2×D4—41no—
202C2×D42−1+414yes—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—24no—
207C22×C4—24no—
208C22×C4—24no—
209C22×C4—24no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—21no—
216C22×C4—42no—
217C22×C4—41no—
218C22×C4—41no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—48no—
222C22×C4—81no—
223C22×C4—81no—
224C22×C4—82no—
225C22×C4—82no—
226C22×C4—81no—
227C22×C4—81no—
228C22×C4—81no—
229C22×C4—81no—
230C22×C4C2×C22⋊C412yes—
231C22×C4C2×C22⋊C411yescharacteristic
232C22×C4C2×C22⋊C412yes—
233C22×C4C2×C22⋊C412yes—
234C22×C4C2×C22⋊C412yes—
235C22×C4C2×C22⋊C412yes—
236C22×C4C23×C411yescharacteristic
237C4⋊C4—24no—
238C4⋊C4—24no—
239C4⋊C4—24no—
240C4⋊C4—22no—
241C4⋊C4—42no—
242C4⋊C4—42no—
243C4⋊C4—41no—
244C4⋊C4—41no—
245C4⋊C4—42no—
246C4⋊C4—44no—
247C4⋊C4—42no—
248C4⋊C4—44no—
249C4⋊C4—44no—
250C4⋊C4—82no—
251C4⋊C4—82no—
252C4⋊C4—82no—
253C4⋊C4—82no—
254C4⋊C4—164no—
255C4⋊C4#9914yes—
256C22⋊C4—24no—
257C22⋊C4—42no—
258C22⋊C4—42no—
259C22⋊C4—42no—
260C22⋊C4—42no—
261C22⋊C4—48no—
262C22⋊C4—48no—
263C22⋊C4—82no—
264C22⋊C4—82no—
265C22⋊C4—82no—
266C22⋊C4—82no—
267C42—24no—
268C42—42no—
269C42—42no—
270C42—41no—
271C42—41no—
272C42—42no—
273C42—42no—
274C42—42no—
275C42—164no—
276C23—21no—
277C23—21no—
278C23—22no—
279C23—22no—
280C23—41no—
281C23—41no—
282C23—42no—
283C23—42no—
284C23#41012yes—
285C23C22×C22⋊C411yescharacteristic
286C23#53614yes—
287Q8—44no—
288Q8—164no—
289D4—44no—
290D4—48no—
291C2×C4—28no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—42no—
303C2×C4—48no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—42no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—42no—
323C2×C4—84no—
324C2×C4—84no—
325C2×C4—84no—
326C2×C4—88no—
327C2×C4—88no—
328C2×C4—88no—
329C2×C4—161no—
330C2×C4—161no—
331C2×C4—162no—
332C2×C4C2×#9914yes—
333C2×C4C2×#9912yes—
334C2×C4#51912yes—
335C22—21no—
336C22—21no—
337C22—22no—
338C22—24no—
339C22—42no—
340C22—44no—
341C22—42no—
342C22—42no—
343C22—41no—
344C22—42no—
345C22—41no—
346C22—82no—
347C22#231011yescharacteristic
348C22C22×#9911yescharacteristic
349C22C2×#41011yescharacteristic
350C4—22no—
351C4—22no—
352C4—24no—
353C4—42no—
354C4—42no—
355C4—42no—
356C4—42no—
357C4—82no—
358C4—81no—
359C4—81no—
360C4—81no—
361C4—82no—
362C4—81no—
363C4—168no—
364C2—21no—
365C2—21no—
366C2—44no—
367C2—42no—
368C2#2182511yescharacteristic
369C2#3336111yescharacteristic
370C2#3336211yescharacteristic
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. #13299C2semidirect
7. C2×#544C4semidirect
9. #2568C4semidirect
11. #2565C4semidirect
17. #2317C4semidirect
21. #2296C4semidirect
22. C2×#382C4semidirect

Maximal subgroups

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

subgroupindex
#356202
#207252
#133002
#132992
#128592
#128572