Finite Group Databasebeta

← #491388

Subgroups — #491388

Order 512.

449 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33774C211yesmaximal, characteristic, complemented
2#33508C211yesmaximal, characteristic, complemented
3#19353C211yesmaximal, characteristic
4#19223C211yesmaximal, characteristic, complemented
5#12479C211yesmaximal, characteristic, complemented
6#12475C211yesmaximal, characteristic, complemented
7#12471C211yesmaximal, characteristic, complemented
8#12467C211yesmaximal, characteristic, complemented
9#12444C211yesmaximal, characteristic, complemented
10#12441C211yesmaximal, characteristic, complemented
11#12435C211yesmaximal, characteristic, complemented
12#12432C211yesmaximal, characteristic, complemented
13#12035C211yesmaximal, characteristic, complemented
14#11984C211yesmaximal, characteristic, complemented
15C2×#1361C211yesmaximal, characteristic, complemented
16#3248—21no—
17#3217—21no—
18#3170C2211yescharacteristic
19C2×#493C2211yescharacteristic
20#2966C2211yescharacteristic
21#2248C2211yescharacteristic
22#2248C2211yescharacteristic
23#2240C2211yescharacteristic
24#2240C2211yescharacteristic
25#2207—21no—
26#2207—21no—
27#2187—21no—
28#2187—21no—
29#2183C2211yescharacteristic, complemented
30Q8×C16C2211yescharacteristic, complemented
31#2169C2211yescharacteristic, complemented
32#2168—21no—
33D4×C16—21no—
34D4×C16C2211yescharacteristic, complemented
35#2164—21no—
36#2162C2211yescharacteristic
37#2151—21no—
38C2×C4⋊C16C2211yescharacteristic
39#1700—21no—
40#1699—21no—
41#1688—21no—
42#1687—21no—
43#1674C2211yescharacteristic, complemented
44#1673C2211yescharacteristic, complemented
45#1670C2211yescharacteristic, complemented
46#1669C2211yescharacteristic, complemented
47#1668C2211yescharacteristic
48#1668C2211yescharacteristic
49#1667C2211yescharacteristic
50#1667C2211yescharacteristic
51#1568C2211yescharacteristic
52C2×#335C2211yescharacteristic
53#1555—21no—
54#1552—21no—
55#1372C2211yescharacteristic, complemented
56#1372C2211yescharacteristic, complemented
57#1371C2211yescharacteristic, complemented
58#1371C2211yescharacteristic, complemented
59#1364C2211yescharacteristic, complemented
60#1364C2211yescharacteristic, complemented
61#1362C2211yescharacteristic, complemented
62#1362C2211yescharacteristic, complemented
63#1361C2211yescharacteristic
64#1361C2211yescharacteristic
65#1361C2211yescharacteristic
66#1361C2211yescharacteristic
67C22×D8—21no—
68C22×D8—21no—
69#533—21no—
70C2×#127—21no—
71#529D411yescharacteristic
72C2×SD32—21no—
73C2×SD32—21no—
74C2×SD32—21no—
75C2×SD32—21no—
76C2×M32—21no—
77C22×C16—21no—
78C22×C16—21no—
79#500C2311yescharacteristic
80#496—41no—
81#494C2311yescharacteristic
82#493—21no—
83#493—21no—
84#493—21no—
85#493C2311yescharacteristic
86#493C2311yescharacteristic
87#490—41no—
88#487—21no—
89#479—41no—
90#475—21no—
91#474—21no—
92#467—21no—
93#467—21no—
94#467—41no—
95#465—41no—
96#464—41no—
97#462—21no—
98#461—21no—
99#460—21no—
100#460—21no—
101#459—21no—
102#459—21no—
103#450—41no—
104#447—41no—
105#446D411yescharacteristic
106C8×Q8C2311yescharacteristic
107C4×Q16—21no—
108C4×Q16—21no—
109#436D411yescharacteristic
110#435—21no—
111C8×D4—21no—
112C8×D4C2311yescharacteristic
113#433—21no—
114#432—21no—
115#432D411yescharacteristic
116C2×C4⋊C8—21no—
117#405—21no—
118C2×C4×C8C2311yescharacteristic
119C4⋊C16—21no—
120C4⋊C16—21no—
121C4⋊C16—21no—
122C4⋊C16D411yescharacteristic
123C4⋊C16D411yescharacteristic
124C4⋊C16C2311yescharacteristic
125C4⋊C16C2311yescharacteristic
126#357—21no—
127#357—21no—
128#357—21no—
129#357—21no—
130C22⋊C16—21no—
131C22⋊C16—21no—
132C22⋊C16D411yescharacteristic
133C22⋊C16D411yescharacteristic
134#346—21no—
135C4×C16—21no—
136C4×C16C2311yescharacteristic
137C4×C16C2311yescharacteristic
138#335—21no—
139#335C2311yescharacteristic
140#335C2311yescharacteristic
141#335C2311yescharacteristic
142#335C2311yescharacteristic
143#334—21no—
144#332—21no—
145#332—21no—
146#332—21no—
147#332—21no—
148#331—21no—
149#331—21no—
150#331—21no—
151#331—21no—
152C22×D4—41no—
153C22×D4—41no—
154C22×D4—41no—
155C2×Q16—21no—
156C2×Q16—21no—
157C2×Q16—41no—
158C2×SD16—41no—
159C2×SD16—41no—
160C2×SD16—41no—
161C2×D8—21no—
162C2×D8—21no—
163C2×D8—21no—
164C2×D8—21no—
165C2×D8—41no—
166C2×D8—41no—
167C2×D8—41no—
168C2×D8—41no—
169C2×D8—81no—
170C2×D8—81no—
171C2×D8—82no—
172C2×M16—21no—
173C22×C8—21no—
174C22×C8—41no—
175C22×C8C2×D411yescharacteristic
176C22×C8#4111yescharacteristic
177C4⋊Q8—21no—
178C4⋊Q8—21no—
179#127—21no—
180#127—21no—
181#127—41no—
182#127—41no—
183#127—41no—
184#126—21no—
185#125—41no—
186#125D811yescharacteristic
187#124D811yescharacteristic
188#123—21no—
189C22⋊Q8—41no—
190C22⋊Q8—41no—
191C22⋊Q8D811yescharacteristic
192#121D811yescharacteristic
193C4×Q8—41no—
194C4×Q8—41no—
195C4×Q8C2×D411yescharacteristic
196C4×D4—21no—
197C4×D4C2×D411yescharacteristic
198#117—21no—
199#117—41no—
200C2×C42C2×D411yescharacteristic
201SD32—81no—
202SD32—81no—
203SD32—81no—
204SD32—81no—
205M32—81no—
206C2×C16—21no—
207C2×C16—21no—
208C2×C16—81no—
209C2×C16—81no—
210C2×C16C2×D411yescharacteristic
211C2×C16C2×D411yescharacteristic
212C2×C16C2×D411yescharacteristic
213C2×C16C2×D411yescharacteristic
214#107—21no—
215#107—21no—
216#107—41no—
217#106—41no—
218C4⋊C8—21no—
219C4⋊C8—21no—
220C4⋊C8—21no—
221C4⋊C8—21no—
222C4⋊C8—21no—
223C4⋊C8—21no—
224C4⋊C8—41no—
225C4⋊C8C2×D411yescharacteristic
226C4⋊C8C2×D411yescharacteristic
227#103—41no—
228#103—41no—
229#103—41no—
230#103—41no—
231#103—41no—
232#102—41no—
233#102—41no—
234#102—41no—
235#102—41no—
236#102—41no—
237C22⋊C8—21no—
238C22⋊C8—21no—
239C22⋊C8—21no—
240C22⋊C8—41no—
241#97—21no—
242#97—21no—
243#97—41no—
244C4×C8—21no—
245C4×C8C2×D411yescharacteristic
246C4×C8C2×D411yescharacteristic
247C4×C8#4111yescharacteristic
248C4×C8C2411yescharacteristic
249C24—81no—
250C2×Q8—41no—
251C2×Q8—41no—
252C2×Q8C2×D811yescharacteristic
253C2×D4—21no—
254C2×D4—41no—
255C2×D4—41no—
256C2×D4—41no—
257C2×D4—41no—
258C2×D4—41no—
259C2×D4—41no—
260C2×D4—82no—
261C2×D4—82no—
262C2×D4—81no—
263C2×D4—81no—
264C2×D4—81no—
265C2×D4—161no—
266C2×D4—161no—
267C2×D4—162no—
268C2×D4C2×D811yescharacteristic
269C22×C4—21no—
270C22×C4#12111yescharacteristic
271C22×C4C2×D811yescharacteristic
272C22×C4#13511yescharacteristic
273Q16—41no—
274Q16—41no—
275Q16—81no—
276Q16—161no—
277SD16—161no—
278SD16—161no—
279SD16—161no—
280D8—41no—
281D8—41no—
282D8—41no—
283D8—41no—
284D8—81no—
285D8—81no—
286D8—82no—
287D8—82no—
288D8—161no—
289D8—161no—
290M16—81no—
291C2×C8—21no—
292C2×C8—21no—
293C2×C8—21no—
294C2×C8—21no—
295C2×C8—41no—
296C2×C8—41no—
297C2×C8—41no—
298C2×C8—41no—
299C2×C8—41no—
300C2×C8—41no—
301C2×C8—41no—
302C2×C8—161no—
303C2×C8#12111yescharacteristic
304C2×C8#12111yescharacteristic
305C2×C8C22×D411yescharacteristic
306C2×C8C2×#4111yescharacteristic
307C4⋊C4—21no—
308C4⋊C4—21no—
309C4⋊C4—21no—
310C4⋊C4—21no—
311C4⋊C4—41no—
312C4⋊C4—41no—
313C4⋊C4—41no—
314C4⋊C4—41no—
315C4⋊C4—81no—
316C4⋊C4—81no—
317C4⋊C4C2×D811yescharacteristic
318C4⋊C4C2×D811yescharacteristic
319C22⋊C4—21no—
320C22⋊C4—21no—
321C22⋊C4—21no—
322C22⋊C4—21no—
323C22⋊C4—21no—
324C22⋊C4—81no—
325C42—21no—
326C42—81no—
327C42#12111yescharacteristic
328C42C2×D811yescharacteristic
329C42#13511yescharacteristic
330C42C22×D411yescharacteristic
331C16—21no—
332C16—21no—
333C16—21no—
334C16—21no—
335C16—41no—
336C16—41no—
337C23—21no—
338C23—81no—
339C23—81no—
340C23—162no—
341C23—162no—
342C23—162no—
343C23#46611yescharacteristic
344Q8—21no—
345Q8—21no—
346Q8—81no—
347Q8—81no—
348Q8—81no—
349Q8—81no—
350D4—21no—
351D4—21no—
352D4—42no—
353D4—81no—
354D4—81no—
355D4—81no—
356D4—81no—
357D4—81no—
358D4—81no—
359D4—81no—
360D4—81no—
361D4—161no—
362D4—161no—
363D4—162no—
364D4—162no—
365D4—162no—
366C2×C4—22no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—81no—
377C2×C4—81no—
378C2×C4#46611yescharacteristic
379C2×C4#46611yescharacteristic
380C2×C4#46611yescharacteristic
381C2×C4C2×#12111yescharacteristic
382C2×C4C22×D811yescharacteristic
383C2×C4C2×#13511yescharacteristic
384C8—21no—
385C8—21no—
386C8—21no—
387C8—41no—
388C8—41no—
389C8—81no—
390C8—81no—
391C8—81no—
392C8—81no—
393C8#57611yescharacteristic
394C8#57711yescharacteristic
395C22—21no—
396C22—21no—
397C22—41no—
398C22—41no—
399C22—41no—
400C22—161no—
401C22—161no—
402C22—161no—
403C22—161no—
404C22—161no—
405C22—161no—
406C22—324no—
407C22#222312yes—
408C22C2×#46611yescharacteristic
409C4—21no—
410C4—21no—
411C4—21no—
412C4—21no—
413C4—41no—
414C4—41no—
415C4—41no—
416C4—161no—
417C4—161no—
418C4#304511yescharacteristic
419C4#306811yescharacteristic
420C4#341511yescharacteristic
421C4#341611yescharacteristic
422C2—22no—
423C2—81no—
424C2—321no—
425C2—321no—
426C2#1941011yescharacteristic
427C2#3361211yescharacteristic
428C2C2×#222311yescharacteristic
subgroupG/H
center, socleC22C2×#466
commutator, Frattini, cosocleC4×C8C24
prime radical#19353C2
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. #33774C2semidirect
2. #33508C2semidirect
4. #19223C2semidirect
5. #12479C2semidirect
6. #12475C2semidirect
7. #12471C2semidirect
8. #12467C2semidirect
9. #12444C2semidirect
10. #12441C2semidirect
11. #12435C2semidirect
12. #12432C2semidirect
13. #12035C2semidirect
14. #11984C2semidirect
15. C2×#1361C2semidirect
29. #2183C22semidirect
30. Q8×C16C22semidirect
31. #2169C22semidirect
34. D4×C16C22semidirect
43. #1674C22semidirect
44. #1673C22semidirect
45. #1670C22semidirect
46. #1669C22semidirect
55. #1372C22semidirect
56. #1372C22semidirect
57. #1371C22semidirect
58. #1371C22semidirect
59. #1364C22semidirect
60. #1364C22semidirect
61. #1362C22semidirect
62. #1362C22semidirect

Maximal subgroups

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

subgroupindex
#337742
#335082
#193532
#192232
#124792
#124752
#124712
#124672
#124442
#124412
#124352
#124322
#120352
#119842
C2×#13612