Finite Group Databasebeta

← #480893

Subgroups — #480893

Order 512.

903 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#38952C211yesmaximal, characteristic
2#17426C212yesmaximal, complemented
3#11567C212yesmaximal, complemented
4#11549C214yesmaximal, complemented
5#11532C211yesmaximal, characteristic, complemented
6#11508C212yesmaximal, complemented
7#11478C211yesmaximal, characteristic, complemented
8#11432C212yesmaximal, complemented
9C2×#554C2211yescharacteristic
10C2×#550C411yescharacteristic
11#2974—22no—
12#2858C411yescharacteristic
13#2672—24no—
14#2620C412yes—
15#2567C411yescharacteristic
16#2548C2211yescharacteristic
17#2501C2212yes—
18#2454—22no—
19#2445—22no—
20#2432C412yes—
21#2311C2212yes—
22C4×C4⋊Q8C2211yescharacteristic
23#2293C411yescharacteristic
24#1472—24no—
25#1470—22no—
26#1467C2212yes—
27#1463C2214yes—
28#1461C2212yes—
29C2×#324C2214yes—
30#1325C2212yes—
31#1325C2212yes—
32#1319C2212yes—
33#1319C2211yescharacteristic
34#1319C2211yescharacteristic
35#1318C2214yes—
36#1318C2214yes—
37C22×#41C2×C412yes—
38#554—42no—
39#554—42no—
40#550—42no—
41#550—42no—
42C2×C4⋊Q8C2×C411yescharacteristic
43C2×C4⋊Q8C2×C411yescharacteristic
44C2×#127—21no—
45C2×#127C2×C411yescharacteristic
46C2×#125D412yes—
47C2×#124—22no—
48C2×#124C2×C412yes—
49C2×#123—24no—
50C2×C22⋊Q8C2×C412yes—
51C2×C22⋊Q8D414yes—
52C2×#121—22no—
53C2×#121C2×C412yes—
54C2×C4×Q8C2×C411yescharacteristic
55C2×C4×Q8C2311yescharacteristic
56C2×C4×D4—21no—
57C2×C4×D4C2×C411yescharacteristic
58C2×C4×D4D412yes—
59C2×#117C2×C412yes—
60C22×C4⋊C4C2312yes—
61#436—42no—
62#435—44no—
63C8×D4—42no—
64#432—22no—
65#431—22no—
66#407—24no—
67C2×C22⋊C8—24no—
68#399—24no—
69#396—24no—
70#395—22no—
71#394—22no—
72#394—24no—
73#394—24no—
74#391C2×C411yescharacteristic
75#391C2×C412yes—
76#391C2311yescharacteristic
77#390—21no—
78#390—22no—
79#390—22no—
80#390C2×C412yes—
81#390C2×C412yes—
82#390C2312yes—
83#389—22no—
84#389—21no—
85#389—22no—
86#388—22no—
87#386—24no—
88#385—24no—
89#384C2×C412yes—
90#384C2311yescharacteristic
91#380—24no—
92C4×C4⋊C4—22no—
93C4×C4⋊C4C2×C411yescharacteristic
94C4×C22⋊C4C2×C412yes—
95C43C2×C411yescharacteristic
96#324—24no—
97#324—22no—
98#324—24no—
99#324—22no—
100#324C2312yes—
101#324C2314yes—
102#324C2312yes—
103C2×#41—24no—
104C2×#41—42no—
105C2×#41—44no—
106C2×#41—82no—
107C22×Q8C22⋊C412yes—
108C22×Q8C22×C411yescharacteristic
109C22×Q8C22×C411yescharacteristic
110C22×D4—22no—
111C22×D4—22no—
112C22×D4—21no—
113C22×D4—21no—
114C22×D4C22⋊C411yescharacteristic
115C22×D4C22×C412yes—
116C23×C4—22no—
117C23×C4C22×C412yes—
118C23×C4C2×D414yes—
119C2×M16—44no—
120C22×C8—44no—
121C4⋊Q8—22no—
122C4⋊Q8—41no—
123C4⋊Q8—41no—
124C4⋊Q8—41no—
125#127—41no—
126#127—41no—
127#127—42no—
128#127—42no—
129#125—42no—
130#125—42no—
131#124—84no—
132#124—82no—
133#123—88no—
134C22⋊Q8—44no—
135C22⋊Q8—44no—
136C22⋊Q8—44no—
137#121—48no—
138#121—44no—
139C4×Q8—41no—
140C4×Q8—41no—
141C4×Q8—41no—
142C4×Q8—41no—
143C4×D4—22no—
144C4×D4—22no—
145C4×D4—22no—
146C4×D4—24no—
147C4×D4—22no—
148C4×D4—42no—
149C4×D4—42no—
150#117—82no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—24no—
157C2×C4⋊C4—44no—
158C2×C4⋊C4C22⋊C411yescharacteristic
159C2×C4⋊C4C22⋊C412yes—
160C2×C4⋊C4C22⋊C414yes—
161C2×C4⋊C4C22×C411yescharacteristic
162C2×C4⋊C4C22×C412yes—
163C2×C4⋊C4C2×D414yes—
164C2×C4⋊C4C2×D412yes—
165C2×C4⋊C4C2411yescharacteristic
166C2×C22⋊C4—22no—
167C2×C22⋊C4—22no—
168C2×C22⋊C4—24no—
169C2×C22⋊C4—22no—
170C2×C22⋊C4—24no—
171C2×C22⋊C4—24no—
172C2×C22⋊C4—24no—
173C2×C22⋊C4C22⋊C412yes—
174C2×C22⋊C4C22⋊C414yes—
175C2×C42—21no—
176C2×C42—21no—
177C2×C42—21no—
178C2×C42C22×C411yescharacteristic
179C2×C42C22×C411yescharacteristic
180C2×C42C22×C411yescharacteristic
181C2×C42C22×C412yes—
182C2×C42C2×D412yes—
183C4⋊C8—42no—
184C4⋊C8—42no—
185C22⋊C8—22no—
186C22⋊C8—24no—
187C22⋊C8—22no—
188C22⋊C8—44no—
189C22⋊C8—44no—
190#97—42no—
191C4×C8—42no—
192#95—22no—
193#95—22no—
194#95—21no—
195#95—24no—
196#95—22no—
197C24—22no—
198C24C2×C22⋊C412yes—
199#41—88no—
200#41—88no—
201C2×Q8—21no—
202C2×Q8—21no—
203C2×Q8—21no—
204C2×Q8—41no—
205C2×Q8—41no—
206C2×Q8—42no—
207C2×Q8—41no—
208C2×Q8—41no—
209C2×Q8—82no—
210C2×Q8#13112yes—
211C2×D4—22no—
212C2×D4—22no—
213C2×D4—42no—
214C2×D4—44no—
215C2×D4—42no—
216C2×D4—42no—
217C2×D4—44no—
218C2×D4—42no—
219C2×D4—44no—
220C2×D4—44no—
221C2×D4—41no—
222C2×D4—41no—
223C2×D4—82no—
224C2×D4—84no—
225C2×D4—82no—
226C2×D4#9912yes—
227C2×D4#9912yes—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—24no—
231C22×C4—24no—
232C22×C4—24no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—21no—
236C22×C4—22no—
237C22×C4—21no—
238C22×C4—42no—
239C22×C4—44no—
240C22×C4—44no—
241C22×C4—82no—
242C22×C4—84no—
243C22×C4C2×C22⋊C412yes—
244C22×C4C2×C22⋊C412yes—
245C22×C4C2×C22⋊C411yescharacteristic
246C22×C4C2×C22⋊C411yescharacteristic
247C22×C4C2×C22⋊C414yes—
248C22×C4C23×C411yescharacteristic
249C22×C4C22×D412yes—
250M16—164no—
251C2×C8—42no—
252C2×C8—42no—
253C2×C8—44no—
254C2×C8—164no—
255C4⋊C4—21no—
256C4⋊C4—21no—
257C4⋊C4—22no—
258C4⋊C4—22no—
259C4⋊C4—41no—
260C4⋊C4—42no—
261C4⋊C4—44no—
262C4⋊C4—44no—
263C4⋊C4—42no—
264C4⋊C4—44no—
265C4⋊C4—41no—
266C4⋊C4—42no—
267C4⋊C4—42no—
268C4⋊C4—44no—
269C4⋊C4—82no—
270C4⋊C4—84no—
271C22⋊C4—24no—
272C22⋊C4—24no—
273C22⋊C4—24no—
274C22⋊C4—44no—
275C22⋊C4—44no—
276C22⋊C4—48no—
277C22⋊C4—84no—
278C22⋊C4—82no—
279C22⋊C4—84no—
280C22⋊C4—84no—
281C42—22no—
282C42—22no—
283C42—21no—
284C42—21no—
285C42—21no—
286C42—21no—
287C42—41no—
288C42—42no—
289C42—42no—
290C42—42no—
291C42—41no—
292C42—41no—
293C23—22no—
294C23—22no—
295C23—42no—
296C23—42no—
297C23—44no—
298C23—44no—
299C23—42no—
300C23—42no—
301C23—42no—
302C23C2×#9914yes—
303C23C22×C22⋊C411yescharacteristic
304Q8—42no—
305Q8—42no—
306Q8—81no—
307Q8—81no—
308Q8—162no—
309D4—24no—
310D4—42no—
311D4—82no—
312D4—82no—
313D4—82no—
314D4—88no—
315D4—82no—
316D4—162no—
317D4—164no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—21no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—24no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—44no—
333C2×C4—44no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—44no—
340C2×C4—42no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—41no—
347C2×C4—42no—
348C2×C4—41no—
349C2×C4—84no—
350C2×C4—84no—
351C2×C4—88no—
352C2×C4C2×#9911yescharacteristic
353C2×C4C2×#9911yescharacteristic
354C2×C4C2×#13111yescharacteristic
355C2×C4#56811yescharacteristic
356C8—82no—
357C8—84no—
358C8—82no—
359C22—21no—
360C22—21no—
361C22—24no—
362C22—24no—
363C22—24no—
364C22—82no—
365C22—82no—
366C22—84no—
367C22—84no—
368C22—82no—
369C22—82no—
370C22—82no—
371C22#288011yescharacteristic
372C22C22×#9911yescharacteristic
373C22#288811yescharacteristic
374C4—21no—
375C4—21no—
376C4—21no—
377C4—42no—
378C4—42no—
379C4—42no—
380C4—42no—
381C4—42no—
382C4—44no—
383C4—41no—
384C4—81no—
385C4—82no—
386C4—81no—
387C4#213312yes—
388C2—21no—
389C2—21no—
390C2—44no—
391C2—162no—
392C2#1741111yescharacteristic
393C2#3335911yescharacteristic
394C2C2×#213311yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C4⋊C4C24
prime radicalC2×#550C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×D4 ◁ C2×#554 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #17426C2semidirect
3. #11567C2semidirect
4. #11549C2semidirect
5. #11532C2semidirect
6. #11508C2semidirect
7. #11478C2semidirect
8. #11432C2semidirect

Maximal subgroups

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

subgroupindex
#389522
#174262
#115672
#115492
#115322
#115082
#114782
#114322