Finite Group Databasebeta

← #480361

Subgroups — #480361

Order 512.

1119 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34275C211yesmaximal, characteristic
2#17369C212yesmaximal, complemented
3#11535C214yesmaximal, complemented
4#11529C214yesmaximal, complemented
5#11450C212yesmaximal, complemented
6#11431C212yesmaximal, complemented
7C2×#535C411yescharacteristic
8C2×#520C411yescharacteristic
9C2×#518C2211yescharacteristic
10#2975—22no—
11#2363—22no—
12#2361—22no—
13#2358—24no—
14#2328C412yes—
15#2326C2212yes—
16#2317C2211yescharacteristic
17C4×#127C2212yes—
18C4×#124C412yes—
19#2297C2211yescharacteristic
20C2×#390C411yescharacteristic
21C4×#117C411yescharacteristic
22#1465C2218yes—
23#1462—24no—
24#1460—22no—
25#1458C2214yes—
26#1325C2218yescomplemented
27#1319C2214yescomplemented
28#1318C2214yescomplemented
29C22×#41C2×C411yescharacteristic
30C23×D4C2×C411yescharacteristic
31#535—24no—
32#535—24no—
33C2×#127C2×C412yes—
34C2×#124C2×C412yes—
35C2×#121—22no—
36C2×#120—22no—
37#520—44no—
38#518—44no—
39C2×C4×Q8C2×C412yes—
40C2×C4×D4—24no—
41C2×C4×D4—22no—
42C2×C4×D4C2×C412yes—
43C2×C4×D4C2×C412yes—
44C2×C4×D4D412yes—
45C2×C4×D4D412yes—
46C2×C4×D4C2312yes—
47C2×#117—21no—
48C2×#117C2×C411yescharacteristic
49C2×#117C2×C411yescharacteristic
50C2×#117C2311yescharacteristic
51C22×C4⋊C4—21no—
52C22×C4⋊C4C2×C411yescharacteristic
53C22×C22⋊C4—22no—
54C22×C22⋊C4D414yes—
55C22×C42C2×C411yescharacteristic
56#436—48no—
57#431—24no—
58#407—28no—
59#391C2×C411yescharacteristic
60#391C2311yescharacteristic
61#390—22no—
62#390—24no—
63#390C2×C412yes—
64#390C2×C412yes—
65#390C2×C411yescharacteristic
66#390C2311yescharacteristic
67#389—22no—
68#388—24no—
69#388—24no—
70#386—24no—
71#386—24no—
72#386—22no—
73#385—24no—
74#385—22no—
75#384C2312yes—
76#382C2×C412yes—
77C4×C4⋊C4—21no—
78C4×C4⋊C4—21no—
79C4×C22⋊C4—24no—
80C4×C22⋊C4—22no—
81C4×C22⋊C4—22no—
82#376C2×C412yes—
83C43C2×C412yes—
84#324C2318yes—
85#323—24no—
86#323—28no—
87C25C22⋊C412yes—
88C2×#41—22no—
89C2×#41—42no—
90C2×#41—42no—
91C2×#41M1614yescomplemented
92C22×Q8C22×C411yescharacteristic
93C22×D4—21no—
94C22×D4—24no—
95C22×D4—22no—
96C22×D4—22no—
97C22×D4—42no—
98C22×D4—42no—
99C22×D4C22⋊C412yes—
100C22×D4C22×C412yes—
101C22×D4C22×C411yescharacteristic
102C23×C4—21no—
103C23×C4—21no—
104C23×C4—21no—
105C23×C4—22no—
106C23×C4C22⋊C414yes—
107C23×C4C22×C411yescharacteristic
108C23×C4C2×D414yes—
109C2×M16—48no—
110#127—24no—
111#127—24no—
112#124—24no—
113#124—24no—
114#121—216no—
115#120—216no—
116C4×Q8—44no—
117C4×D4—28no—
118C4×D4—44no—
119C4×D4—44no—
120C4×D4—44no—
121C4×D4—44no—
122C4×D4—48no—
123C4×D4—88no—
124#117—22no—
125#117—24no—
126#117—42no—
127#117—44no—
128#117M1614yescomplemented
129C2×C4⋊C4—21no—
130C2×C4⋊C4—21no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—24no—
133C2×C4⋊C4—44no—
134C2×C4⋊C4—42no—
135C2×C4⋊C4C22⋊C411yescharacteristic
136C2×C4⋊C4C22⋊C411yescharacteristic
137C2×C4⋊C4C22×C412yes—
138C2×C4⋊C4C22×C411yescharacteristic
139C2×C4⋊C4C2411yescharacteristic
140C2×C22⋊C4—21no—
141C2×C22⋊C4—24no—
142C2×C22⋊C4—24no—
143C2×C22⋊C4—22no—
144C2×C22⋊C4—22no—
145C2×C22⋊C4—24no—
146C2×C22⋊C4—22no—
147C2×C22⋊C4—48no—
148C2×C22⋊C4—84no—
149C2×C22⋊C4C22⋊C412yes—
150C2×C22⋊C4C22⋊C414yes—
151C2×C22⋊C4C2×D412yes—
152C2×C22⋊C4C2×D414yes—
153C2×C42—22no—
154C2×C42—22no—
155C2×C42—21no—
156C2×C42—22no—
157C2×C42—42no—
158C2×C42C22×C412yes—
159C2×C42C22×C412yes—
160C2×C42C22×C411yescharacteristic
161C2×C42C22×C411yescharacteristic
162C2×C42C2×D412yes—
163C4⋊C8—44no—
164C22⋊C8—28no—
165C22⋊C8—48no—
166C4×C8—44no—
167#95—22no—
168#95—22no—
169#95—22no—
170#95—24no—
171#95—22no—
172C24—21no—
173C24—24no—
174C24—42no—
175C24—42no—
176C24—82no—
177C24—82no—
178C24C2×C22⋊C414yes—
179C24C2×C22⋊C411yescharacteristic
180#41—48no—
181#41—48no—
182C2×Q8—21no—
183C2×Q8—41no—
184C2×Q8—41no—
185C2×Q8C2×M1612yes—
186C2×D4—24no—
187C2×D4—216no—
188C2×D4—22no—
189C2×D4—22no—
190C2×D4—21no—
191C2×D4—24no—
192C2×D4—24no—
193C2×D4—24no—
194C2×D4—42no—
195C2×D4—42no—
196C2×D4—41no—
197C2×D4—41no—
198C2×D4—48no—
199C2×D4—48no—
200C2×D4—42no—
201C2×D4—42no—
202C2×D4—84no—
203C2×D4—84no—
204C2×D4C2×M1612yes—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—24no—
211C22×C4—28no—
212C22×C4—21no—
213C22×C4—22no—
214C22×C4—44no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—44no—
221C22×C4—81no—
222C22×C4—84no—
223C22×C4—82no—
224C22×C4—84no—
225C22×C4—84no—
226C22×C4C2×C22⋊C412yes—
227C22×C4C2×C22⋊C412yes—
228C22×C4C2×C22⋊C412yes—
229C22×C4C2×C22⋊C411yescharacteristic
230C22×C4C2×M1612yes—
231C22×C4C2×M1612yes—
232C22×C4C23×C411yescharacteristic
233C22×C4C22×D412yes—
234M16—168no—
235C2×C8—48no—
236C4⋊C4—24no—
237C4⋊C4—22no—
238C4⋊C4—41no—
239C4⋊C4—41no—
240C4⋊C4—42no—
241C4⋊C4—42no—
242C4⋊C4—44no—
243C4⋊C4—84no—
244C4⋊C4#9914yes—
245C22⋊C4—216no—
246C22⋊C4—24no—
247C22⋊C4—24no—
248C22⋊C4—28no—
249C22⋊C4—42no—
250C22⋊C4—44no—
251C22⋊C4—48no—
252C22⋊C4—44no—
253C22⋊C4—44no—
254C22⋊C4—82no—
255C22⋊C4—84no—
256C42—22no—
257C42—24no—
258C42—24no—
259C42—41no—
260C42—41no—
261C42—44no—
262C42—44no—
263C42—42no—
264C42—82no—
265C42C2×M1614yes—
266C23—21no—
267C23—21no—
268C23—24no—
269C23—22no—
270C23—24no—
271C23—24no—
272C23—41no—
273C23—41no—
274C23—44no—
275C23—44no—
276C23—42no—
277C23—42no—
278C23—48no—
279C23—41no—
280C23—41no—
281C23—88no—
282C23—162no—
283C23—162no—
284C23—164no—
285C23#40712yes—
286C23C22×C22⋊C411yescharacteristic
287Q8—42no—
288Q8—42no—
289D4—42no—
290D4—44no—
291D4—44no—
292D4—42no—
293D4—48no—
294D4—48no—
295D4—816no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—24no—
299C2×C4—24no—
300C2×C4—22no—
301C2×C4—24no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—41no—
309C2×C4—42no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—48no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—44no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—42no—
321C2×C4—41no—
322C2×C4—88no—
323C2×C4—84no—
324C2×C4—164no—
325C2×C4—164no—
326C2×C4—162no—
327C2×C4#40712yes—
328C2×C4C2×#9914yes—
329C2×C4C2×#9912yes—
330C2×C4#41312yes—
331C2×C4C22×M1611yescharacteristic
332C2×C4C22×M1611yescharacteristic
333C8—88no—
334C22—21no—
335C22—21no—
336C22—22no—
337C22—42no—
338C22—42no—
339C22—44no—
340C22—48no—
341C22—41no—
342C22—41no—
343C22—44no—
344C22—44no—
345C22—42no—
346C22—44no—
347C22—84no—
348C22—82no—
349C22—81no—
350C22—81no—
351C22—168no—
352C22C2×#40711yescharacteristic
353C22C22×#9911yescharacteristic
354C22C2×#41311yescharacteristic
355C4—24no—
356C4—21no—
357C4—21no—
358C4—42no—
359C4—44no—
360C4—42no—
361C4—42no—
362C4—42no—
363C4—42no—
364C4—81no—
365C4—82no—
366C4#146514yes—
367C2—21no—
368C2—21no—
369C2—42no—
370C2—84no—
371C2—82no—
372C2C2×#146511yescharacteristic
373C2#3335911yescharacteristic
374C2#3337111yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C4⋊C4C24
prime radicalC2×#535C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C4⋊C4 ◁ C2×#518 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #17369C2semidirect
3. #11535C2semidirect
4. #11529C2semidirect
5. #11450C2semidirect
6. #11431C2semidirect
26. #1325C22semidirect
27. #1319C22semidirect
28. #1318C22semidirect
91. C2×#41M16semidirect
128. #117M16semidirect

Maximal subgroups

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

subgroupindex
#342752
#173692
#115352
#115292
#114502
#114312