Finite Group Databasebeta

← #6407967

Subgroups — #6407967

Order 512.

1749 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#52294C211yesmaximal, characteristic
2#23349C216yesmaximal, complemented
3#23348C213yesmaximal, complemented
4#23310C213yesmaximal, complemented
5#23308C213yesmaximal, complemented
6#21635C213yesmaximal, complemented
7C8×#124C216yesmaximal, complemented
8#21442C213yesmaximal, complemented
9C8×#120C213yesmaximal, complemented
10C22×#131C2211yescharacteristic, complemented
11C24×C8C2211yescharacteristic
12C2×C8×Q8C2213yescomplemented
13C2×C8×D4C2216yes—
14C2×C8×D4C2213yes—
15#2763C413yes—
16#2753C416yes—
17#2736C411yescharacteristic
18#2712C413yes—
19#2670C413yes—
20#2398C2213yes—
21#2388C2213yes—
22C4×#124C2216yes—
23C4×#120C2213yes—
24#1985C2213yescomplemented
25#1984C2216yescomplemented
26#1955C2213yes—
27#1954C2216yes—
28#1924C2216yes—
29#1924C2216yescomplemented
30#1923C2216yes—
31#1923C2216yes—
32#1923C2216yescomplemented
33#1923C2216yescomplemented
34#1870C2213yes—
35#1870C2216yes—
36#1869C2213yes—
37#1869C2216yes—
38#1851C2216yescomplemented
39#1851C2213yescomplemented
40#1850C2216yes—
41#1850C2213yes—
42#1832C2213yescomplemented
43#1831C2216yescomplemented
44#1780C2213yescomplemented
45#1779C2213yes—
46C8×C22⋊C4C2213yes—
47C8×C22⋊C4C2216yes—
48C8×C22⋊C4C2213yescomplemented
49C8×C22⋊C4C2216yescomplemented
50#1726C2211yescharacteristic, complemented
51C42×C8C2212yescomplemented
52C22×#41C2311yescharacteristic
53C24×C4C2311yescharacteristic
54C2×#131—46no—
55C22×M16C2313yes—
56C23×C8—46no—
57C23×C8C2313yes—
58C23×C8C2313yes—
59C23×C8C2313yes—
60C2×#124C2×C416yes—
61C2×#120C2×C413yes—
62C2×C4×Q8C2313yes—
63C2×C4×D4C2313yes—
64C2×C4×D4C2316yes—
65C8×Q8—46no—
66C8×D4—412no—
67C8×D4—46no—
68C2×C4⋊C8C2316yes—
69C2×C4⋊C8C2316yes—
70C2×C4⋊C8C2313yes—
71C2×C4⋊C8C2313yes—
72C2×C22⋊C8C2316yes—
73C2×C22⋊C8C2313yes—
74C2×C22⋊C8C2316yes—
75C2×C22⋊C8C2313yes—
76C2×C22⋊C8C2313yes—
77C2×C22⋊C8C2316yes—
78C2×C22⋊C8C2313yes—
79C2×C22⋊C8C2316yes—
80C2×#97C2313yes—
81C2×C4×C8C2316yes—
82C2×C4×C8C2313yes—
83C2×C4×C8C2316yes—
84#400C2×C416yes—
85#397C2×C413yes—
86#397C2×C413yes—
87#395C2×C413yes—
88#394C2×C416yes—
89#394C2×C413yes—
90#394C2×C416yes—
91#393C2×C413yes—
92#393C2×C413yes—
93#393C2×C413yes—
94#392C2×C413yes—
95#392C2×C416yes—
96#391C2×C416yes—
97#390C2×C416yes—
98#388C2×C416yes—
99#388C2×C416yes—
100#388C2×C416yes—
101#388C2×C416yes—
102#386C2×C416yes—
103#386C2×C413yes—
104#385C2×C413yes—
105#385C2×C416yes—
106#382C2×C416yes—
107#379C2×C413yes—
108C4×C22⋊C4C2316yes—
109C4×C22⋊C4C2313yes—
110C4×C22⋊C4C2316yes—
111C4×C22⋊C4C2313yes—
112#376C2311yescharacteristic
113C43C2312yes—
114#336C2316yes—
115#336C2316yes—
116#336C2316yes—
117#336C2316yes—
118#336C2316yes—
119#336C2316yes—
120C25C22×C411yescharacteristic
121C2×#41—43no—
122C2×#41—43no—
123C22×Q8C22×C413yes—
124C22×Q8C22×C411yescharacteristic
125C22×D4C22×C416yes—
126C22×D4C22×C413yes—
127C22×D4C22×C413yes—
128C23×C4—43no—
129C23×C4—43no—
130C23×C4C22×C413yes—
131C23×C4C2413yes—
132C23×C4C2413yes—
133#131—416no—
134C2×M16—43no—
135C2×M16—46no—
136C22×C8—212no—
137C22×C8—26no—
138C22×C8—212no—
139C22×C8—26no—
140C22×C8—43no—
141C22×C8—46no—
142C22×C8—416no—
143C22×C8C2413yes—
144C22×C8C2413yes—
145C22×C8C2416yes—
146C22×C8C2413yes—
147C22×C8C2411yescharacteristic
148#124—224no—
149#120—212no—
150C4×Q8—43no—
151C4×Q8—43no—
152C4×D4—46no—
153C4×D4—43no—
154C4×D4—43no—
155C4×D4—46no—
156C2×C4⋊C4C22×C416yes—
157C2×C4⋊C4C22×C413yes—
158C2×C4⋊C4C22×C416yes—
159C2×C4⋊C4C22×C413yes—
160C2×C22⋊C4C22×C416yes—
161C2×C22⋊C4C22×C416yes—
162C2×C22⋊C4C22×C416yes—
163C2×C22⋊C4C22×C416yes—
164C2×C22⋊C4C22×C416yes—
165C2×C22⋊C4C22×C413yes—
166C2×C22⋊C4C22×C416yes—
167C2×C22⋊C4C22×C413yes—
168C2×C22⋊C4C22×C413yes—
169C2×C22⋊C4C22×C416yes—
170C2×C22⋊C4C22×C413yes—
171C2×C42C22×C416yes—
172C2×C42C22×C412yes—
173C2×C42C2416yes—
174C2×C42C2413yes—
175C4⋊C8—46no—
176C4⋊C8—43no—
177C4⋊C8—46no—
178C4⋊C8—43no—
179C22⋊C8—43no—
180C22⋊C8—43no—
181C22⋊C8—46no—
182C22⋊C8—46no—
183C22⋊C8—46no—
184C22⋊C8—43no—
185C22⋊C8—46no—
186C22⋊C8—43no—
187#97—43no—
188C4×C8—212no—
189C4×C8—26no—
190C4×C8—46no—
191#95C22×C411yescharacteristic
192#95C22×C413yes—
193#95C22×C413yes—
194#95C22×C416yes—
195#95C22×C416yes—
196#95C22×C416yes—
197#95C22×C416yes—
198#95C22×C413yes—
199#95C22×C416yes—
200C24—26no—
201C24—43no—
202C24—41no—
203C24C23×C413yes—
204C24C23×C413yes—
205#41—44no—
206#41—412no—
207C2×Q8—212no—
208C2×Q8—43no—
209C2×Q8—43no—
210C2×D4—26no—
211C2×D4—212no—
212C2×D4—212no—
213C2×D4—46no—
214C2×D4—43no—
215C2×D4—43no—
216C2×D4—43no—
217C2×D4—46no—
218C22×C4—26no—
219C22×C4—23no—
220C22×C4—26no—
221C22×C4—23no—
222C22×C4—26no—
223C22×C4—23no—
224C22×C4—26no—
225C22×C4—23no—
226C22×C4—46no—
227C22×C4—43no—
228C22×C4—44no—
229C22×C4—412no—
230C22×C4C23×C413yes—
231C22×C4C23×C413yes—
232C22×C4C23×C413yes—
233C22×C4C23×C413yes—
234C22×C4C23×C416yes—
235C22×C4C23×C416yes—
236C22×C4C2511yescharacteristic
237M16—412no—
238C2×C8—26no—
239C2×C8—23no—
240C2×C8—26no—
241C2×C8—23no—
242C2×C8—26no—
243C2×C8—23no—
244C2×C8—26no—
245C2×C8—26no—
246C2×C8—26no—
247C2×C8—412no—
248C2×C8—412no—
249C2×C8—412no—
250C2×C8H(5,𝔽2)16yes—
251C4⋊C4—46no—
252C4⋊C4—43no—
253C4⋊C4—46no—
254C4⋊C4—43no—
255C22⋊C4—212no—
256C22⋊C4—212no—
257C22⋊C4—212no—
258C22⋊C4—212no—
259C22⋊C4—43no—
260C22⋊C4—46no—
261C22⋊C4—46no—
262C22⋊C4—46no—
263C22⋊C4—43no—
264C22⋊C4—43no—
265C22⋊C4#131112yescomplemented
266C42—212no—
267C42—26no—
268C42—23no—
269C42—26no—
270C42—23no—
271C42—42no—
272C23—26no—
273C23—23no—
274C23—26no—
275C23—23no—
276C23—26no—
277C23—23no—
278C23—26no—
279C23—23no—
280C23—23no—
281C23—412no—
282C23—412no—
283C23—44no—
284C23C2×#13116yes—
285C23C24×C411yescharacteristic
286Q8—412no—
287Q8—44no—
288D4—424no—
289D4—412no—
290D4—412no—
291C2×C4—26no—
292C2×C4—26no—
293C2×C4—26no—
294C2×C4—23no—
295C2×C4—23no—
296C2×C4—26no—
297C2×C4—23no—
298C2×C4—26no—
299C2×C4—23no—
300C2×C4—26no—
301C2×C4—23no—
302C2×C4—26no—
303C2×C4—26no—
304C2×C4—23no—
305C2×C4—412no—
306C2×C4—43no—
307C2×C4—43no—
308C2×C4—46no—
309C2×C4—43no—
310C2×C4—46no—
311C2×C4—43no—
312C2×C4C2×#131112yes—
313C2×C4C2×H(5,𝔽2)13yes—
314C2×C4#58513yes—
315C8—43no—
316C8—43no—
317C8—46no—
318C8—43no—
319C8H(3,𝔽4)14yes—
320C22—212no—
321C22—26no—
322C22—416no—
323C22—43no—
324C22—43no—
325C22—46no—
326C22—43no—
327C22—46no—
328C22—43no—
329C22—43no—
330C22—46no—
331C22C4×H(5,𝔽2)13yes—
332C22C22×#13113yes—
333C22#357311yescharacteristic
334C4—212no—
335C4—43no—
336C4—43no—
337C4—43no—
338C4—46no—
339C4—43no—
340C4C2×H(3,𝔽4)11yescharacteristic
341C4#352913yes—
342C2—46no—
343C2—83no—
344C2C4×H(3,𝔽4)11yescharacteristic
345C2#6030513yes—
346C2#6030613yes—
subgroupG/H
centerC22×C8C24
commutator, socleC23C24×C4
Frattini, cosocleC22×C4C25
prime radical#52294C2
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C22×C8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #23349C2semidirect
3. #23348C2semidirect
4. #23310C2semidirect
5. #23308C2semidirect
6. #21635C2semidirect
7. C8×#124C2semidirect
8. #21442C2semidirect
9. C8×#120C2semidirect
10. C22×#131C22semidirect
12. C2×C8×Q8C22semidirect
24. #1985C22semidirect
25. #1984C22semidirect
29. #1924C22semidirect
32. #1923C22semidirect
33. #1923C22semidirect
38. #1851C22semidirect
39. #1851C22semidirect
42. #1832C22semidirect
43. #1831C22semidirect
44. #1780C22semidirect
48. C8×C22⋊C4C22semidirect
49. C8×C22⋊C4C22semidirect
50. #1726C22semidirect
51. C42×C8C22semidirect
265. C22⋊C4#131semidirect

Maximal subgroups

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

subgroupindex
#522942
#233492
#233482
#233102
#233082
#216352
C8×#1242
#214422
C8×#1202