Finite Group Databasebeta

← #491417

Subgroups — #491417

Order 512.

387 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33796C211yesmaximal, characteristic, complemented
2#33507C211yesmaximal, characteristic
3#19359C211yesmaximal, characteristic
4#19220C211yesmaximal, characteristic
5#12481C211yesmaximal, characteristic, complemented
6#12477C211yesmaximal, characteristic, complemented
7#12473C211yesmaximal, characteristic, complemented
8#12469C211yesmaximal, characteristic, complemented
9#12461C211yesmaximal, characteristic, complemented
10#12458C211yesmaximal, characteristic, complemented
11#12455C211yesmaximal, characteristic, complemented
12#12452C211yesmaximal, characteristic, complemented
13#12034C211yesmaximal, characteristic, complemented
14#11987C211yesmaximal, characteristic, complemented
15#11941C211yesmaximal, characteristic, complemented
16#3238—21no—
17#3233—21no—
18#3170C2211yescharacteristic
19C2×#500C2211yescharacteristic
20#2966C2211yescharacteristic
21#2252C2211yescharacteristic
22#2252C2211yescharacteristic
23#2241C2211yescharacteristic
24#2241C2211yescharacteristic
25#2236—21no—
26#2236—21no—
27#2227—21no—
28#2227—21no—
29#2183C2211yescharacteristic
30Q8×C16C2211yescharacteristic
31#2169—21no—
32#2169C2211yescharacteristic
33#2168—21no—
34D4×C16C2211yescharacteristic
35#2164—21no—
36#2162C2211yescharacteristic
37#2151—21no—
38#2150C2211yescharacteristic
39#1704—21no—
40#1703—21no—
41#1696—21no—
42#1695—21no—
43#1676C2211yescharacteristic
44#1676C2211yescharacteristic
45#1676C2211yescharacteristic
46#1675C2211yescharacteristic
47#1675C2211yescharacteristic
48#1675C2211yescharacteristic
49#1670C2211yescharacteristic
50#1669C2211yescharacteristic
51#1568C2211yescharacteristic
52#1564C2211yescharacteristic
53#1558—21no—
54#1554—21no—
55#1372C2211yescharacteristic
56#1372C2211yescharacteristic
57#1371C2211yescharacteristic
58#1371C2211yescharacteristic
59#1365C2211yescharacteristic
60#1365C2211yescharacteristic
61#1365C2211yescharacteristic
62#1365C2211yescharacteristic
63#1365C2211yescharacteristic
64#1365C2211yescharacteristic
65#1362C2211yescharacteristic
66#1362C2211yescharacteristic
67#533—21no—
68C2×C4⋊Q8—21no—
69#529D411yescharacteristic
70C2×M32—21no—
71C2×M32—21no—
72C22×C16—21no—
73#501—41no—
74#500—21no—
75#500—21no—
76#500—21no—
77#500C2311yescharacteristic
78#500C2311yescharacteristic
79#500C2311yescharacteristic
80#499—41no—
81#494C2311yescharacteristic
82#489—41no—
83#488—41no—
84#487—21no—
85#485—41no—
86#483—41no—
87#482—41no—
88#480—41no—
89#477—21no—
90#477—21no—
91#477—21no—
92#476—21no—
93#476—21no—
94#476—21no—
95#475—21no—
96#474—21no—
97#467—21no—
98#467—21no—
99#446D411yescharacteristic
100C8×Q8C2311yescharacteristic
101C4×Q16—21no—
102C4×Q16—21no—
103#436D411yescharacteristic
104#435—21no—
105C8×D4—21no—
106C8×D4C2311yescharacteristic
107#433—21no—
108#432—21no—
109#432D411yescharacteristic
110#431—21no—
111C2×#107—21no—
112C2×#107—21no—
113#405—21no—
114C2×C4×C8C2311yescharacteristic
115#366—21no—
116#366—21no—
117#366—21no—
118#366—21no—
119C4⋊C16—21no—
120C4⋊C16—21no—
121C4⋊C16—21no—
122C4⋊C16D411yescharacteristic
123C4⋊C16D411yescharacteristic
124C4⋊C16C2311yescharacteristic
125C4⋊C16C2311yescharacteristic
126#358—21no—
127#358—21no—
128#358—21no—
129#358—21no—
130C22⋊C16—21no—
131C22⋊C16—21no—
132C22⋊C16D411yescharacteristic
133C22⋊C16D411yescharacteristic
134#346—21no—
135C4×C16—21no—
136C4×C16C2311yescharacteristic
137C4×C16C2311yescharacteristic
138#335C2311yescharacteristic
139#335C2311yescharacteristic
140#335C2311yescharacteristic
141#335C2311yescharacteristic
142#334—21no—
143#334—21no—
144#333—21no—
145#333—21no—
146#333—21no—
147#333—21no—
148#333—21no—
149#333—21no—
150#332—21no—
151#332—21no—
152C22×Q8—41no—
153C2×Q16—21no—
154C2×Q16—21no—
155C2×M16—21no—
156C22×C8—21no—
157C22×C8C2×D411yescharacteristic
158C22×C8#4111yescharacteristic
159C4⋊Q8—21no—
160C4⋊Q8—21no—
161C4⋊Q8—21no—
162C4⋊Q8—21no—
163C4⋊Q8—41no—
164C4⋊Q8—41no—
165C4⋊Q8—41no—
166#126—21no—
167#125—41no—
168#125D811yescharacteristic
169#124D811yescharacteristic
170#123—21no—
171C22⋊Q8—41no—
172C22⋊Q8—41no—
173C22⋊Q8D811yescharacteristic
174#121D811yescharacteristic
175C4×Q8—41no—
176C4×Q8—41no—
177C4×Q8C2×D411yescharacteristic
178C4×D4—21no—
179C4×D4C2×D411yescharacteristic
180#117—21no—
181#117—41no—
182C2×C4⋊C4—41no—
183C2×C4⋊C4—41no—
184C2×C42C2×D411yescharacteristic
185M32—81no—
186M32—81no—
187C2×C16—21no—
188C2×C16—21no—
189C2×C16—81no—
190C2×C16C2×D411yescharacteristic
191C2×C16C2×D411yescharacteristic
192C2×C16C2×D411yescharacteristic
193C2×C16C2×D411yescharacteristic
194#107—21no—
195#107—21no—
196#107—21no—
197#107—21no—
198#107—21no—
199#107—21no—
200#107—41no—
201#107—41no—
202#107—41no—
203#107—41no—
204#107—41no—
205#107—41no—
206#106—41no—
207#106—41no—
208#106—41no—
209#106—41no—
210C4⋊C8—21no—
211C4⋊C8—21no—
212C4⋊C8—21no—
213C4⋊C8—21no—
214C4⋊C8—21no—
215C4⋊C8—21no—
216C4⋊C8C2×D411yescharacteristic
217C4⋊C8C2×D411yescharacteristic
218#103—41no—
219#103—41no—
220#103—41no—
221#103—41no—
222#103—41no—
223#103—41no—
224#102—41no—
225#102—41no—
226#102—41no—
227#102—41no—
228C22⋊C8—21no—
229C22⋊C8—21no—
230C22⋊C8—21no—
231C22⋊C8—41no—
232C22⋊C8—41no—
233#97—21no—
234#97—21no—
235#97—41no—
236C4×C8—21no—
237C4×C8—41no—
238C4×C8C2×D411yescharacteristic
239C4×C8C2×D411yescharacteristic
240C4×C8#4111yescharacteristic
241C4×C8C2411yescharacteristic
242C2×Q8—41no—
243C2×Q8—41no—
244C2×Q8—41no—
245C2×Q8—41no—
246C2×Q8—81no—
247C2×Q8—162no—
248C2×Q8C2×D811yescharacteristic
249C2×D4—21no—
250C2×D4C2×D811yescharacteristic
251C22×C4—21no—
252C22×C4—81no—
253C22×C4#12111yescharacteristic
254C22×C4C2×D811yescharacteristic
255C22×C4#13511yescharacteristic
256Q16—41no—
257Q16—41no—
258Q16—81no—
259M16—81no—
260C2×C8—21no—
261C2×C8—21no—
262C2×C8—21no—
263C2×C8—21no—
264C2×C8—41no—
265C2×C8—41no—
266C2×C8—41no—
267C2×C8—41no—
268C2×C8—41no—
269C2×C8—41no—
270C2×C8—41no—
271C2×C8#12111yescharacteristic
272C2×C8#12111yescharacteristic
273C2×C8C22×D411yescharacteristic
274C2×C8C2×#4111yescharacteristic
275C4⋊C4—21no—
276C4⋊C4—21no—
277C4⋊C4—21no—
278C4⋊C4—21no—
279C4⋊C4—41no—
280C4⋊C4—41no—
281C4⋊C4—41no—
282C4⋊C4—41no—
283C4⋊C4—41no—
284C4⋊C4—41no—
285C4⋊C4—41no—
286C4⋊C4—41no—
287C4⋊C4—81no—
288C4⋊C4—81no—
289C4⋊C4—81no—
290C4⋊C4—81no—
291C4⋊C4C2×D811yescharacteristic
292C4⋊C4C2×D811yescharacteristic
293C22⋊C4—21no—
294C22⋊C4—21no—
295C22⋊C4—21no—
296C22⋊C4—21no—
297C22⋊C4—21no—
298C22⋊C4—81no—
299C42—21no—
300C42—81no—
301C42#12111yescharacteristic
302C42C2×D811yescharacteristic
303C42#13511yescharacteristic
304C42C22×D411yescharacteristic
305C16—21no—
306C16—21no—
307C16—21no—
308C16—21no—
309C16—41no—
310C16—41no—
311C23—21no—
312C23#46611yescharacteristic
313Q8—21no—
314Q8—21no—
315Q8—81no—
316Q8—81no—
317Q8—161no—
318Q8—161no—
319Q8—161no—
320Q8—162no—
321D4—21no—
322D4—21no—
323D4—81no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—81no—
335C2×C4—81no—
336C2×C4—81no—
337C2×C4—81no—
338C2×C4—162no—
339C2×C4#46611yescharacteristic
340C2×C4#46611yescharacteristic
341C2×C4#46611yescharacteristic
342C2×C4C2×#12111yescharacteristic
343C2×C4C22×D811yescharacteristic
344C2×C4C2×#13511yescharacteristic
345C8—21no—
346C8—21no—
347C8—21no—
348C8—41no—
349C8—41no—
350C8—81no—
351C8—81no—
352C8—81no—
353C8—81no—
354C8#57711yescharacteristic
355C8#57811yescharacteristic
356C22—21no—
357C22—21no—
358C22—21no—
359C22—41no—
360C22—41no—
361C22—41no—
362C22C2×#46611yescharacteristic
363C4—21no—
364C4—21no—
365C4—21no—
366C4—21no—
367C4—41no—
368C4—41no—
369C4—41no—
370C4—161no—
371C4—161no—
372C4—161no—
373C4—161no—
374C4#304611yescharacteristic
375C4#306911yescharacteristic
376C4#341711yescharacteristic
377C4#341711yescharacteristic
378C2—41no—
379C2—81no—
380C2#1941611yescharacteristic
381C2#3360811yescharacteristic
382C2#3361811yescharacteristic
subgroupG/H
center, socleC22C2×#466
commutator, Frattini, cosocleC4×C8C24
prime radical#121D8
LCSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2966 ◁ G
LECSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #33796C2semidirect
5. #12481C2semidirect
6. #12477C2semidirect
7. #12473C2semidirect
8. #12469C2semidirect
9. #12461C2semidirect
10. #12458C2semidirect
11. #12455C2semidirect
12. #12452C2semidirect
13. #12034C2semidirect
14. #11987C2semidirect
15. #11941C2semidirect

Maximal subgroups

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

subgroupindex
#337962
#335072
#193592
#192202
#124812
#124772
#124732
#124692
#124612
#124582
#124552
#124522
#120342
#119872
#119412