Finite Group Databasebeta

← #480941

Subgroups — #480941

Order 512.

914 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#39174C211yesmaximal, characteristic
2#17419C212yesmaximal
3#11572C214yesmaximal, complemented
4#11570C212yesmaximal, complemented
5#11535C212yesmaximal, complemented
6#11531C211yesmaximal, characteristic, complemented
7#11481C212yesmaximal, complemented
8#11477C211yesmaximal, characteristic, complemented
9C2×#557C411yescharacteristic
10C2×#550C2211yescharacteristic
11#2880—22no—
12#2858C411yescharacteristic
13#2711—24no—
14#2641C412yes—
15#2566C2212yes—
16#2548C412yes—
17#2548C411yescharacteristic
18#2542C2211yescharacteristic
19#2432—22no—
20#2426—22no—
21#2326C2212yes—
22C4×#124C2211yescharacteristic
23#2295C411yescharacteristic
24#1471—22no—
25#1471—24no—
26#1465C2212yes—
27#1465C2214yes—
28#1459C2212yes—
29#1459C2214yes—
30#1325C2214yes—
31#1325C2214yes—
32#1324C2212yes—
33#1324C2212yes—
34#1319C2212yes—
35#1319C2211yescharacteristic
36#1319C2211yescharacteristic
37C22×#41D412yes—
38C2×#131—42no—
39#557—42no—
40#557—42no—
41#550—42no—
42#550—42no—
43C2×C4⋊Q8—21no—
44C2×C4⋊Q8C2×C411yescharacteristic
45C2×#127C2×C411yescharacteristic
46C2×#127C2×C412yes—
47C2×#125—22no—
48C2×#125C2×C412yes—
49C2×#124—22no—
50C2×#124C2×C412yes—
51C2×#124C2×C411yescharacteristic
52C2×C22⋊Q8—22no—
53C2×#121D412yes—
54C2×#121D414yes—
55C2×C4×Q8—21no—
56C2×C4×Q8C2×C412yes—
57C2×C4×Q8C2×C411yescharacteristic
58C2×C4×Q8C2×C411yescharacteristic
59C2×C4×D4C2312yes—
60C2×C4×D4C2311yescharacteristic
61#408—42no—
62#408—44no—
63#407—22no—
64#407—24no—
65C2×C22⋊C8—22no—
66C2×C22⋊C8—24no—
67#399—22no—
68#396—24no—
69#394—24no—
70#394—24no—
71#394—24no—
72#394—22no—
73#391—21no—
74#391C2311yescharacteristic
75#390—22no—
76#390—22no—
77#390—24no—
78#390C2×C412yes—
79#390C2×C412yes—
80#390C2×C412yes—
81#390C2×C411yescharacteristic
82#390C2×C412yes—
83#389—21no—
84#388—24no—
85#388—22no—
86#387—22no—
87#385—24no—
88#385—24no—
89#384C2312yes—
90#382C2311yescharacteristic
91C4×C4⋊C4C2×C412yes—
92C4×C4⋊C4C2×C411yescharacteristic
93C4×C22⋊C4—22no—
94#376C2×C412yes—
95C43C2×C411yescharacteristic
96#324C2312yes—
97#324C2314yes—
98#324C2312yes—
99#323—24no—
100#323—22no—
101#323—24no—
102#323—22no—
103C2×#41—22no—
104C2×#41—22no—
105C2×#41—24no—
106C2×#41—84no—
107C22×Q8—21no—
108C22×Q8C22⋊C411yescharacteristic
109C22×Q8C22×C411yescharacteristic
110C22×D4C22⋊C412yes—
111C22×D4C22⋊C414yes—
112C22×D4C22⋊C414yes—
113C22×D4C22⋊C412yes—
114C22×D4C22×C412yes—
115C22×D4C22×C411yescharacteristic
116C23×C4C2×D412yes—
117C23×C4C2×D414yes—
118#131—88no—
119C2×M16—44no—
120C2×M16—42no—
121C22×C8—44no—
122C22×C8—42no—
123C4⋊Q8—41no—
124C4⋊Q8—41no—
125C4⋊Q8—42no—
126C4⋊Q8—42no—
127#127—44no—
128#127—41no—
129#127—41no—
130#125—44no—
131#125—48no—
132#124—24no—
133#124—22no—
134#124—42no—
135#124—41no—
136#124—84no—
137C22⋊Q8—84no—
138#121—42no—
139#121—42no—
140#121—44no—
141#121—44no—
142C4×Q8—22no—
143C4×Q8—22no—
144C4×Q8—44no—
145C4×Q8—41no—
146C4×Q8—41no—
147C4×Q8—42no—
148C4×Q8—42no—
149C4×D4—44no—
150C4×D4—41no—
151C4×D4—41no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—24no—
160C2×C4⋊C4C22⋊C411yescharacteristic
161C2×C4⋊C4C22⋊C412yes—
162C2×C4⋊C4C22×C411yescharacteristic
163C2×C4⋊C4C22×C412yes—
164C2×C4⋊C4C2411yescharacteristic
165C2×C22⋊C4—22no—
166C2×C22⋊C4—24no—
167C2×C22⋊C4—24no—
168C2×C22⋊C4—24no—
169C2×C22⋊C4—24no—
170C2×C22⋊C4—22no—
171C2×C22⋊C4—24no—
172C2×C22⋊C4C2×D414yes—
173C2×C22⋊C4C2×D412yes—
174C2×C42—22no—
175C2×C42—21no—
176C2×C42—21no—
177C2×C42—21no—
178C2×C42C22×C411yescharacteristic
179C2×C42C22×C412yes—
180C2×C42C22×C412yes—
181C2×C42C22×C411yescharacteristic
182C2×C42C22×C411yescharacteristic
183C22⋊C8—22no—
184C22⋊C8—24no—
185C22⋊C8—22no—
186C22⋊C8—42no—
187C22⋊C8—44no—
188C22⋊C8—42no—
189C22⋊C8—44no—
190#95—22no—
191#95—22no—
192#95—22no—
193#95—21no—
194#95—22no—
195#95—22no—
196C24C2×C22⋊C412yes—
197C24C2×C22⋊C414yes—
198#41—48no—
199#41—84no—
200#41—88no—
201C2×Q8—21no—
202C2×Q8—42no—
203C2×Q8—44no—
204C2×Q8—41no—
205C2×Q8—41no—
206C2×Q8—41no—
207C2×Q8—41no—
208C2×Q8#9912yes—
209C2×Q8#9912yes—
210C2×Q8#13112yescomplemented
211C2×D4—22no—
212C2×D4—21no—
213C2×D4—22no—
214C2×D4—21no—
215C2×D4—24no—
216C2×D4—22no—
217C2×D4—22no—
218C2×D4—24no—
219C2×D4—44no—
220C2×D4—42no—
221C2×D4—41no—
222C2×D4—42no—
223C2×D4—41no—
224C2×D4—44no—
225C2×D4—42no—
226C2×D4—84no—
227C2×D4—82no—
228C2×D4—84no—
229C22×C4—21no—
230C22×C4—24no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—24no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—22no—
237C22×C4—21no—
238C22×C4—22no—
239C22×C4—44no—
240C22×C4—84no—
241C22×C4C2×C22⋊C411yescharacteristic
242C22×C4C2×C22⋊C412yes—
243C22×C4C2×C22⋊C411yescharacteristic
244C22×C4C2×C22⋊C412yes—
245C22×C4C23×C411yescharacteristic
246C22×C4C22×D412yes—
247M16—88no—
248M16—84no—
249C2×C8—42no—
250C2×C8—44no—
251C2×C8—42no—
252C2×C8—88no—
253C2×C8—84no—
254C4⋊C4—21no—
255C4⋊C4—21no—
256C4⋊C4—24no—
257C4⋊C4—41no—
258C4⋊C4—41no—
259C4⋊C4—42no—
260C4⋊C4—44no—
261C4⋊C4—42no—
262C4⋊C4—42no—
263C4⋊C4—42no—
264C4⋊C4—44no—
265C4⋊C4—44no—
266C4⋊C4—82no—
267C4⋊C4—84no—
268C22⋊C4—44no—
269C22⋊C4—42no—
270C22⋊C4—44no—
271C22⋊C4—48no—
272C22⋊C4—82no—
273C22⋊C4—84no—
274C22⋊C4—84no—
275C22⋊C4—84no—
276C22⋊C4—84no—
277C42—24no—
278C42—21no—
279C42—21no—
280C42—21no—
281C42—22no—
282C42—21no—
283C42—22no—
284C42—41no—
285C42—44no—
286C42—42no—
287C42—41no—
288C42—41no—
289C42—81no—
290C23—24no—
291C23—22no—
292C23—22no—
293C23—24no—
294C23—22no—
295C23—24no—
296C23—42no—
297C23—44no—
298C23—42no—
299C23—44no—
300C23C22×C22⋊C411yescharacteristic
301Q8—24no—
302Q8—42no—
303Q8—42no—
304Q8—42no—
305Q8—82no—
306Q8—82no—
307D4—48no—
308D4—44no—
309D4—82no—
310D4—81no—
311D4—82no—
312D4—81no—
313D4—82no—
314D4—84no—
315D4—164no—
316D4—162no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—21no—
323C2×C4—22no—
324C2×C4—42no—
325C2×C4—48no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—44no—
329C2×C4—41no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—44no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—41no—
341C2×C4—42no—
342C2×C4—41no—
343C2×C4—42no—
344C2×C4—41no—
345C2×C4—82no—
346C2×C4—84no—
347C2×C4C2×#9911yescharacteristic
348C2×C4C2×#9914yes—
349C2×C4C2×#9911yescharacteristic
350C2×C4C2×#13111yescharacteristic
351C2×C4#56811yescharacteristic
352C8—82no—
353C8—84no—
354C8—82no—
355C22—21no—
356C22—21no—
357C22—44no—
358C22—42no—
359C22—44no—
360C22—42no—
361C22—44no—
362C22—42no—
363C22—82no—
364C22—84no—
365C22—82no—
366C22—84no—
367C22#288011yescharacteristic
368C22C22×#9911yescharacteristic
369C22#288911yescharacteristic
370C4—21no—
371C4—21no—
372C4—21no—
373C4—24no—
374C4—42no—
375C4—42no—
376C4—42no—
377C4—41no—
378C4—81no—
379C4—82no—
380C4—81no—
381C4—82no—
382C4—81no—
383C4#212712yes—
384C2—21no—
385C2—21no—
386C2—84no—
387C2—82no—
388C2#1741411yescharacteristic
389C2C2×#212711yescharacteristic
390C2#3337111yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C4⋊C4C24
prime radicalC2×#550C22
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×Q8 ◁ C2×#550 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #11572C2semidirect
4. #11570C2semidirect
5. #11535C2semidirect
6. #11531C2semidirect
7. #11481C2semidirect
8. #11477C2semidirect
210. C2×Q8#131semidirect

Maximal subgroups

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

subgroupindex
#391742
#174192
#115722
#115702
#115352
#115312
#114812
#114772