Finite Group Databasebeta

← #6378895

Subgroups — #6378895

Order 512.

1654 conjugacy classes of subgroups.

The subgroup lattice has 343 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34578C211yesmaximal, characteristic
2#22295C214yesmaximal, complemented
3#22292C218yesmaximal, complemented
4#21934C211yesmaximal, characteristic
5#21933C212yesmaximal
6#21601C218yesmaximal, complemented
7C8×C22⋊Q8C214yesmaximal, complemented
8#20385C212yesmaximal
9C2×C8×C22⋊C4C211yesmaximal, characteristic
10C24×C8C2211yescharacteristic
11C2×C8×Q8C2214yescomplemented
12C2×#407C2212yes—
13C22×C22⋊C8C2211yescharacteristic
14C2×#405C2214yes—
15C22×C4×C8C2212yes—
16#2667C414yes—
17#2665C414yes—
18#2659C412yes—
19#2646C412yes—
20#2626C414yes—
21#2396C2218yes—
22C4×C22⋊Q8C2214yes—
23#2280C2212yes—
24C2×C4×C22⋊C4C2211yescharacteristic
25#1985C2218yescomplemented
26#1984C2214yescomplemented
27#1940C2218yescomplemented
28#1939C2214yescomplemented
29#1917C2218yescomplemented
30#1917C2218yescomplemented
31#1916C2218yescomplemented
32#1851C2218yes—
33#1851C2218yes—
34#1850C2218yes—
35#1798C2218yes—
36#1797C2214yes—
37C8×C4⋊C4C2218yescomplemented
38C8×C4⋊C4C2214yescomplemented
39#1752C2214yes—
40#1752C2214yes—
41C8×C22⋊C4C2214yes—
42C8×C22⋊C4C2214yes—
43C8×C22⋊C4C2214yes—
44C8×C22⋊C4C2214yes—
45#1729C2214yes—
46C2×#336C2212yes—
47C24×C4C2311yescharacteristic
48C22×M16C2314yes—
49C23×C8—44no—
50C23×C8Q818yescomplemented
51C23×C8C2311yescharacteristic
52C23×C8C2312yes—
53C23×C8C2312yes—
54C23×C8C2311yescharacteristic
55C23×C8C2312yes—
56C2×C22⋊Q8C2×C414yes—
57C2×C4×Q8C2314yes—
58C2×#117C2314yes—
59C22×C22⋊C4C2×C412yes—
60C22×C42C2312yes—
61C8×Q8—48no—
62C2×C4⋊C8C2314yes—
63C2×C4⋊C8C2314yes—
64C2×C4⋊C8C2318yes—
65C2×C4⋊C8C2318yes—
66#407—28no—
67C2×C22⋊C8—24no—
68C2×C22⋊C8C2314yes—
69C2×C22⋊C8C2314yes—
70C2×C22⋊C8C2314yes—
71#405—48no—
72C2×#97C2318yes—
73C2×C4×C8—44no—
74C2×C4×C8C2314yes—
75C2×C4×C8C2318yes—
76C2×C4×C8C2314yes—
77#400C2×C418yes—
78#399C2×C418yes—
79#398C2×C418yes—
80#398C2×C418yes—
81#396C2×C418yes—
82#396C2×C418yes—
83#395C2×C418yes—
84#393C2×C418yes—
85#391C2×C414yes—
86#389C2×C414yes—
87#387C2×C418yes—
88#385C2×C418yes—
89#381C2×C414yes—
90#380C2×C414yes—
91#380C2×C414yes—
92#380C2×C412yes—
93#380C2×C412yes—
94#379C2×C414yes—
95C4×C4⋊C4C2318yes—
96C4×C4⋊C4C2314yes—
97C4×C22⋊C4C2×C414yes—
98C4×C22⋊C4C2314yes—
99C4×C22⋊C4C2314yes—
100C4×C22⋊C4C2314yes—
101C2×#95C2×C411yescharacteristic
102C2×#95C2×C411yescharacteristic
103#336C2318yes—
104#336C2318yes—
105#336C2318yes—
106#336C2318yes—
107#336C2318yes—
108#336C2318yes—
109C25C22×C411yescharacteristic
110C22×Q8C22×C414yes—
111C23×C4—42no—
112C23×C4—42no—
113C23×C4C22×C412yes—
114C23×C4C22×C411yescharacteristic
115C23×C4C22×C412yes—
116C23×C4C22×C412yes—
117C23×C4C2×Q814yes—
118C23×C4#4114yes—
119C23×C4C2412yes—
120C23×C4C2411yescharacteristic
121C2×M16—28no—
122C2×M16—44no—
123C2×M16—44no—
124C22×C8—24no—
125C22×C8—28no—
126C22×C8—24no—
127C22×C8—416no—
128C22×C8—44no—
129C22×C8—42no—
130C22×C8—42no—
131C22×C8—42no—
132C22×C8C2×Q814yes—
133C22×C8C2×Q818yes—
134C22×C8C2×Q814yes—
135C22×C8C2×Q818yes—
136C22×C8C2414yes—
137C22×C8C2418yes—
138C22×C8C2411yescharacteristic
139C22×C8C2412yes—
140C22×C8C2411yescharacteristic
141C22⋊Q8—216no—
142C4×Q8—44no—
143C4×Q8—44no—
144#117—44no—
145#117—44no—
146C2×C4⋊C4C22×C418yes—
147C2×C4⋊C4C22×C414yes—
148C2×C4⋊C4C22×C414yes—
149C2×C4⋊C4C22×C414yes—
150C2×C4⋊C4C22×C414yes—
151C2×C4⋊C4C22×C418yes—
152C2×C4⋊C4C22×C414yes—
153C2×C4⋊C4C22×C418yes—
154C2×C22⋊C4—28no—
155C2×C22⋊C4C22×C418yes—
156C2×C22⋊C4C22×C418yes—
157C2×C22⋊C4C22×C418yes—
158C2×C22⋊C4C22×C414yes—
159C2×C22⋊C4C22×C414yes—
160C2×C42—42no—
161C2×C42—42no—
162C2×C42C22×C414yes—
163C2×C42C2418yes—
164C2×C42C2414yes—
165C4⋊C8—44no—
166C4⋊C8—48no—
167C4⋊C8—44no—
168C4⋊C8—48no—
169C22⋊C8—28no—
170C22⋊C8—28no—
171C22⋊C8—28no—
172#97—48no—
173C4×C8—28no—
174C4×C8—216no—
175C4×C8—44no—
176#95C22×C414yes—
177#95C22×C414yes—
178#95C22×C414yes—
179#95C22×C414yes—
180#95C22×C418yes—
181#95C22×C418yes—
182#95C22×C418yes—
183#95C22×C418yes—
184C24—24no—
185C24—42no—
186C24—41no—
187C24—41no—
188C24C4×Q814yes—
189C24C23×C412yes—
190C24C23×C411yescharacteristic
191C2×Q8—28no—
192C2×Q8—44no—
193C2×Q8—44no—
194C22×C4—22no—
195C22×C4—24no—
196C22×C4—22no—
197C22×C4—24no—
198C22×C4—24no—
199C22×C4—44no—
200C22×C4—44no—
201C22×C4—48no—
202C22×C4—42no—
203C22×C4—42no—
204C22×C4—44no—
205C22×C4—42no—
206C22×C4—42no—
207C22×C4—42no—
208C22×C4C4×Q818yes—
209C22×C4C4×Q814yes—
210C22×C4#13118yes—
211C22×C4C23×C414yes—
212C22×C4C23×C418yes—
213C22×C4C23×C418yes—
214C22×C4C23×C411yescharacteristic
215C22×C4C23×C412yes—
216C22×C4C23×C414yes—
217C22×C4C22×Q814yes—
218C22×C4C22×Q812yes—
219C22×C4C2×#4112yes—
220C22×C4C2×#4114yes—
221C22×C4C2511yescharacteristic
222M16—416no—
223C2×C8—22no—
224C2×C8—24no—
225C2×C8—28no—
226C2×C8—216no—
227C2×C8—24no—
228C2×C8—28no—
229C2×C8—24no—
230C2×C8—28no—
231C2×C8—24no—
232C2×C8—28no—
233C2×C8—48no—
234C2×C8—44no—
235C2×C8—48no—
236C2×C8C22×Q812yes—
237C2×C8C22×Q814yes—
238C2×C8C22×Q812yes—
239C2×C8H(5,𝔽2)14yes—
240C4⋊C4—28no—
241C4⋊C4—216no—
242C4⋊C4—44no—
243C4⋊C4—48no—
244C4⋊C4—44no—
245C4⋊C4—44no—
246C4⋊C4—44no—
247C4⋊C4—48no—
248C22⋊C4—216no—
249C22⋊C4—216no—
250C22⋊C4—44no—
251C22⋊C4—44no—
252C22⋊C4—48no—
253C42—24no—
254C42—28no—
255C42—24no—
256C42—28no—
257C42—44no—
258C23—22no—
259C23—21no—
260C23—21no—
261C23—24no—
262C23—28no—
263C23—24no—
264C23—22no—
265C23—22no—
266C23—44no—
267C23—44no—
268C23—48no—
269C23—48no—
270C23C2×C4×Q814yes—
271C23C2×C4×Q812yes—
272C23C2×#13114yes—
273C23C24×C411yescharacteristic
274Q8—416no—
275C2×C4—24no—
276C2×C4—22no—
277C2×C4—24no—
278C2×C4—24no—
279C2×C4—22no—
280C2×C4—24no—
281C2×C4—22no—
282C2×C4—24no—
283C2×C4—24no—
284C2×C4—28no—
285C2×C4—24no—
286C2×C4—28no—
287C2×C4—28no—
288C2×C4—24no—
289C2×C4—28no—
290C2×C4—28no—
291C2×C4—28no—
292C2×C4—28no—
293C2×C4—24no—
294C2×C4—44no—
295C2×C4—48no—
296C2×C4—48no—
297C2×C4—48no—
298C2×C4C2×C4×Q812yes—
299C2×C4C2×C4×Q814yes—
300C2×C4C2×#13118yes—
301C2×C4C23×Q811yescharacteristic
302C2×C4C22×#4111yescharacteristic
303C2×C4C2×H(5,𝔽2)12yes—
304C2×C4#58512yes—
305C8—22no—
306C8—24no—
307C8—44no—
308C8—48no—
309C8#54314yes—
310C22—22no—
311C22—22no—
312C22—24no—
313C22—24no—
314C22—24no—
315C22—24no—
316C22—24no—
317C22—416no—
318C22—44no—
319C22—44no—
320C22#296014yes—
321C22C22×C4×Q811yescharacteristic
322C22C4×H(5,𝔽2)12yes—
323C22C22×#13112yes—
324C22#357311yescharacteristic
325C22#357311yescharacteristic
326C4—22no—
327C4—24no—
328C4—28no—
329C4—48no—
330C4—48no—
331C4—44no—
332C4C2×#54311yescharacteristic
333C4#345811yescharacteristic
334C4#345912yes—
335C2—24no—
336C2—44no—
337C2C4×#54311yescharacteristic
338C2C2×#296011yescharacteristic
339C2#6019111yescharacteristic
340C2#6030412yes—
341C2#6030612yes—
subgroupG/H
centerC22×C8C24
commutator, socleC23C24×C4
Frattini, cosocleC22×C4C25
prime radicalC25C22×C4
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. #22295C2semidirect
3. #22292C2semidirect
6. #21601C2semidirect
7. C8×C22⋊Q8C2semidirect
11. C2×C8×Q8C22semidirect
25. #1985C22semidirect
26. #1984C22semidirect
27. #1940C22semidirect
28. #1939C22semidirect
29. #1917C22semidirect
30. #1917C22semidirect
31. #1916C22semidirect
37. C8×C4⋊C4C22semidirect
38. C8×C4⋊C4C22semidirect
50. C23×C8Q8semidirect

Maximal subgroups

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

subgroupindex
#345782
#222952
#222922
#219342
#219332
#216012
C8×C22⋊Q82
#203852
C2×C8×C22⋊C42