Finite Group Databasebeta

← #7623177

Subgroups — #7623177

Order 512.

1692 conjugacy classes of subgroups.

The subgroup lattice has 327 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60667C211yesmaximal, characteristic, complemented
2#60647C211yesmaximal, characteristic
3#33707C213yesmaximal, complemented
4#33704C213yesmaximal, complemented
5#33702C2112yesmaximal, complemented
6#33697C213yesmaximal, complemented
7C2×#2240C213yesmaximal, complemented
8#33381C214yesmaximal, complemented
9C4×#504C211yesmaximal, characteristic, complemented
10C2×#576—21no—
11C2×#576C2212yes—
12#3503—21no—
13#3498—21no—
14C2×#504C2212yes—
15#3165—23no—
16#3164—23no—
17#3164C2216yescomplemented
18#3160C2213yescomplemented
19C2×#500C2213yescomplemented
20#3155—23no—
21#3155C2216yes—
22#3151—23no—
23#3147C2213yes—
24C2×#493C2213yes—
25#2913—21no—
26#2913C2212yes—
27C4×#131C2211yescharacteristic
28#2244C22112yescomplemented
29#2243C22112yescomplemented
30#2242C22112yescomplemented
31#2241C2211yescharacteristic, complemented
32#2241C2213yescomplemented
33#2240C2213yescomplemented
34#2240C22112yescomplemented
35#2240C2216yescomplemented
36#2240C2213yescomplemented
37#2140C22112yescomplemented
38#2139C22112yescomplemented
39#2138C22112yescomplemented
40C2×#357C22112yescomplemented
41#2108C2216yescomplemented
42C4×M32C2213yescomplemented
43C2×C4×C16C2213yescomplemented
44C2×H(5,𝔽2)—22no—
45#576—216no—
46#576—48no—
47C2×AGL(1,ℤ/8)—26no—
48C2×AGL(1,ℤ/8)—26no—
49C2×#135—23no—
50C2×#135C2316yes—
51C22×D8—23no—
52C22×D8C2316yes—
53C2×#131D411yescharacteristic
54C2×#131C2311yescharacteristic
55Q82—41no—
56#557—41no—
57#554—43no—
58#550—41no—
59#550—41no—
60D4×Q8—43no—
61D42—43no—
62D42—43no—
63#544—26no—
64#535—26no—
65#533—23no—
66#532—23no—
67C2×C4⋊Q8—23no—
68C2×#127—23no—
69#520—22no—
70C4×#41D411yescharacteristic
71#504—28no—
72C2×M32C2316yes—
73C22×C16C2316yes—
74#501—212no—
75#500—23no—
76#500—23no—
77#500C2313yes—
78#500C2316yes—
79#500C2311yescharacteristic
80#499—23no—
81#499C2316yes—
82#498—23no—
83#496—212no—
84#495—23no—
85#495C2316yes—
86#494C2311yescharacteristic
87#493—23no—
88#493—23no—
89#493C2313yes—
90#493C2316yes—
91#492—23no—
92#428—26no—
93#428—26no—
94#427—23no—
95#427C2316yes—
96C2×#107—23no—
97C2×#107C2316yes—
98#405D413yes—
99#405C2313yes—
100C4×M16D413yes—
101C2×C4×C8C2313yes—
102#358C2314yes—
103#358C23112yes—
104#357C23112yes—
105#357C23124yes—
106#357C23112yes—
107#346C2316yes—
108C4×C16C2311yescharacteristic
109C4×C16C2313yes—
110C4×C16C2316yes—
111H(5,𝔽2)—48no—
112H(5,𝔽2)—48no—
113C2×#41—26no—
114C2×#41—42no—
115C2×#41C2×D411yescharacteristic
116C22×Q8—43no—
117C22×D4—26no—
118C22×D4—46no—
119C22×D4—43no—
120C22×D4—43no—
121AGL(1,ℤ/8)—424no—
122AGL(1,ℤ/8)—424no—
123#135—224no—
124#135—412no—
125C2×Q16—21no—
126C2×Q16C2412yes—
127C2×SD16—26no—
128C2×SD16—26no—
129C2×D8—23no—
130C2×D8—26no—
131C2×D8—26no—
132C2×D8—224no—
133C2×D8—412no—
134C2×D8C2416yes—
135#131SD1618yes—
136#131#4118yes—
137C2×M16C2×D413yes—
138C2×M16C2×D413yes—
139C22×C8C2×D413yes—
140C22×C8C2413yes—
141C4⋊Q8—23no—
142C4⋊Q8—26no—
143C4⋊Q8—21no—
144C4⋊Q8—21no—
145C4⋊Q8—46no—
146C4⋊Q8—43no—
147C4⋊Q8—43no—
148#127—23no—
149#127—26no—
150#127—43no—
151#127—43no—
152#125—26no—
153#125—46no—
154#124—26no—
155C22⋊Q8—43no—
156C22⋊Q8—43no—
157C22⋊Q8—412no—
158#121—43no—
159#121—43no—
160#121—46no—
161#121—46no—
162#120—412no—
163C4×Q8—42no—
164C4×Q8—43no—
165C4×Q8—43no—
166C4×Q8D811yescharacteristic
167C4×Q8D811yescharacteristic
168C4×D4—46no—
169C4×D4—43no—
170C4×D4—43no—
171C4×D4D813yes—
172C4×D4D813yes—
173#117—26no—
174#117C2×D413yes—
175C2×C4⋊C4—26no—
176C2×C4⋊C4—43no—
177C2×C42C2×D413yes—
178M32—212no—
179C2×C16—212no—
180C2×C16C2412yes—
181C2×C16C2416yes—
182#107—23no—
183#107—26no—
184#107—26no—
185#107—21no—
186#107C2416yes—
187#107C2412yes—
188#106—26no—
189#106—26no—
190#97C2×D416yes—
191C4×C8C2×D413yes—
192C4×C8C2×D413yes—
193C4×C8C2411yescharacteristic
194C4×C8C2413yes—
195C24—46no—
196#41—412no—
197#41—412no—
198#41—88no—
199#41#12414yes—
200#41C2×SD1614yes—
201C2×Q8—22no—
202C2×Q8—41no—
203C2×Q8—43no—
204C2×Q8—46no—
205C2×Q8—166no—
206C2×Q8C2×D811yescharacteristic
207C2×D4—26no—
208C2×D4—46no—
209C2×D4—412no—
210C2×D4—43no—
211C2×D4—43no—
212C2×D4—46no—
213C2×D4—46no—
214C2×D4—46no—
215C2×D4—412no—
216C2×D4—824no—
217C2×D4—166no—
218C2×D4—166no—
219C2×D4C2×D813yes—
220C22×C4—46no—
221C22×C4—46no—
222C22×C4C2×D813yes—
223C22×C4C2×D813yes—
224C22×C4C22×D413yes—
225Q16—24no—
226Q16—42no—
227SD16—412no—
228SD16—412no—
229D8—212no—
230D8—412no—
231D8—46no—
232D8—412no—
233M16#124112yes—
234M16C2×SD16112yes—
235C2×C8C2×SD16112yes—
236C2×C8C22×D413yes—
237C2×C8C22×D411yescharacteristic
238C2×C8C22×D413yes—
239C2×C8C2×#41112yes—
240C2×C8C2511yescharacteristic
241C4⋊C4—22no—
242C4⋊C4—26no—
243C4⋊C4—46no—
244C4⋊C4—43no—
245C4⋊C4—46no—
246C4⋊C4—41no—
247C4⋊C4—41no—
248C4⋊C4—41no—
249C4⋊C4—46no—
250C4⋊C4—46no—
251C4⋊C4C2×D813yes—
252C4⋊C4C2×D813yes—
253C22⋊C4—46no—
254C22⋊C4—46no—
255C22⋊C4—46no—
256C22⋊C4C2×D816yes—
257C42—46no—
258C42C2×D813yes—
259C42C2×D813yes—
260C42C22×D411yescharacteristic
261C42C22×D413yes—
262C16—22no—
263C16—26no—
264C23—46no—
265C23—812no—
266C23—166no—
267C23—166no—
268C23C22×D813yes—
269Q8—42no—
270Q8—42no—
271Q8—86no—
272Q8—82no—
273Q8—166no—
274Q8#48614yes—
275D4—46no—
276D4—46no—
277D4—812no—
278D4—812no—
279D4—86no—
280D4—86no—
281D4—166no—
282D4—166no—
283D4#486112yes—
284C2×C4—26no—
285C2×C4—26no—
286C2×C4—42no—
287C2×C4—46no—
288C2×C4—42no—
289C2×C4—812no—
290C2×C4—166no—
291C2×C4C2×#12416yes—
292C2×C4C22×D813yes—
293C2×C4C22×D813yes—
294C2×C4C22×D813yes—
295C2×C4C22×D811yescharacteristic
296C2×C4C22×D811yescharacteristic
297C2×C4C22×SD1616yes—
298C2×C4C23×D411yescharacteristic
299C8C2×#12416yes—
300C8C22×SD1612yes—
301C8C22×SD1616yes—
302C8C22×#4112yes—
303C22—23no—
304C22—23no—
305C22—86no—
306C22—86no—
307C22—86no—
308C22—3212no—
309C22C2×#48616yes—
310C22C23×D811yescharacteristic
311C4—23no—
312C4—23no—
313C4—86no—
314C4—82no—
315C4—82no—
316C4C2×#48616yes—
317C4#341512yes—
318C4#341612yes—
319C4C22×#12411yescharacteristic
320C4C23×SD1611yescharacteristic
321C2—26no—
322C2—166no—
323C2C22×#48611yescharacteristic
324C2C2×#341511yescharacteristic
325C2C2×#341611yescharacteristic
subgroupG/H
center, socleC22C23×D8
commutator, Frattini, cosocleC2×C8C25
prime radical#60647C2
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C4×#41 ◁ C4×#131 ◁ G
LECSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #60667C2semidirect
3. #33707C2semidirect
4. #33704C2semidirect
5. #33702C2semidirect
6. #33697C2semidirect
7. C2×#2240C2semidirect
8. #33381C2semidirect
9. C4×#504C2semidirect
17. #3164C22semidirect
18. #3160C22semidirect
19. C2×#500C22semidirect
28. #2244C22semidirect
29. #2243C22semidirect
30. #2242C22semidirect
31. #2241C22semidirect
32. #2241C22semidirect
33. #2240C22semidirect
34. #2240C22semidirect
35. #2240C22semidirect
36. #2240C22semidirect
37. #2140C22semidirect
38. #2139C22semidirect
39. #2138C22semidirect
40. C2×#357C22semidirect
41. #2108C22semidirect
42. C4×M32C22semidirect
43. C2×C4×C16C22semidirect

Maximal subgroups

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

subgroupindex
#606672
#606472
#337072
#337042
#337022
#336972
C2×#22402
#333812
C4×#5042