Finite Group Databasebeta

← #6528518

Subgroups — #6528518

Order 512.

1824 conjugacy classes of subgroups.

The subgroup lattice has 337 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35181C211yesmaximal, characteristic, complemented
2C2×#2542C211yesmaximal, characteristic
3#29110C214yesmaximal
4#27966C214yesmaximal, complemented
5#27822C218yesmaximal, complemented
6#27736C218yesmaximal, complemented
7#21833C212yesmaximal, complemented
8#20732C212yesmaximal
9#20536C211yesmaximal, characteristic, complemented
10C22×C4⋊Q8C2211yescharacteristic
11C22×C22⋊Q8—22no—
12C22×C4×D4C2211yescharacteristic
13C2×#431C2212yes—
14#2851C2214yes—
15#2646—22no—
16#2571C2214yescomplemented
17#2570—24no—
18#2552C2214yescomplemented
19#2542—24no—
20#2542C2218yes—
21C2×#399—22no—
22#2317—22no—
23C2×#390C2212yes—
24#2285C2212yes—
25C2×#384C2212yes—
26C2×C4×C4⋊C4C2211yescharacteristic
27#2100C22116yescomplemented
28#2073C2218yes—
29#2025C2218yes—
30#1992C2218yescomplemented
31#1990C2214yescomplemented
32#1990C2214yescomplemented
33#1988C2218yescomplemented
34#1936C22116yes—
35#1836C2218yescomplemented
36#1823C2214yes—
37#1823C2218yes—
38#1802C2214yes—
39#1802C2214yes—
40#1786C2214yes—
41#1770C2218yescomplemented
42C4×#102C2218yes—
43#1749C2214yes—
44C23×Q8—21no—
45C23×D4C2311yescharacteristic
46C24×C4—21no—
47C2×C4⋊Q8—22no—
48C2×C4⋊Q8—28no—
49C2×C4⋊Q8C2314yes—
50C2×#125C2314yes—
51C2×C22⋊Q8—28no—
52C2×C22⋊Q8—28no—
53C2×C22⋊Q8—416no—
54C2×C4×D4—22no—
55C2×C4×D4—28no—
56C2×C4×D4C2314yes—
57C2×#117—22no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—24no—
60C22×C4⋊C4D412yes—
61C22×C4⋊C4C2311yescharacteristic
62C22×C4⋊C4C2312yes—
63C22×C22⋊C4—21no—
64C22×C22⋊C4—24no—
65C22×C42C2312yes—
66C22×C42C2311yescharacteristic
67#431—28no—
68C2×C4⋊C8C2314yes—
69C2×#102C2318yes—
70C2×#102C2318yes—
71C2×C22⋊C8C2314yes—
72C2×#97C2318yes—
73C2×C4×C8C2314yes—
74#400—28no—
75#399—216no—
76#398—28no—
77#390—24no—
78#390C2318yes—
79#389—24no—
80#389—24no—
81#389C2318yes—
82#387—28no—
83#385—28no—
84#384—24no—
85#384—24no—
86#384C2312yes—
87#384C2312yes—
88#384C2318yes—
89#382—24no—
90#382—22no—
91#382C2314yes—
92#380—22no—
93#380C2314yes—
94C4×C4⋊C4—22no—
95C4×C4⋊C4D414yes—
96C4×C4⋊C4C2314yes—
97C4×C4⋊C4C2314yes—
98C2×#95D412yes—
99#340C2318yes—
100#340C2318yes—
101#340C23116yes—
102#340C23116yes—
103#336C2318yes—
104C25—21no—
105C22×Q8—24no—
106C22×Q8—44no—
107C22×Q8—48no—
108C22×D4—24no—
109C22×D4—21no—
110C22×D4—24no—
111C22×D4—42no—
112C22×D4C2412yes—
113C23×C4—22no—
114C23×C4—21no—
115C23×C4—24no—
116C23×C4—44no—
117C23×C4—48no—
118C23×C4C2×D414yes—
119C23×C4C2×D411yescharacteristic
120C23×C4C2×D411yescharacteristic
121C23×C4C2411yescharacteristic
122C22×C8C2418yes—
123C4⋊Q8—28no—
124C4⋊Q8—28no—
125C4⋊Q8—48no—
126#125—48no—
127C22⋊Q8—432no—
128C22⋊Q8—432no—
129C4×D4—28no—
130C4×D4—28no—
131C4×D4—48no—
132#117—48no—
133C2×C4⋊C4—28no—
134C2×C4⋊C4—24no—
135C2×C4⋊C4—28no—
136C2×C4⋊C4—216no—
137C2×C4⋊C4—24no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—28no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—28no—
146C2×C4⋊C4—416no—
147C2×C4⋊C4—44no—
148C2×C4⋊C4C2×D414yes—
149C2×C4⋊C4C2×D414yes—
150C2×C4⋊C4C2412yes—
151C2×C4⋊C4C2414yes—
152C2×C4⋊C4C2414yes—
153C2×C4⋊C4C2414yes—
154C2×C22⋊C4—24no—
155C2×C22⋊C4—24no—
156C2×C22⋊C4—216no—
157C2×C22⋊C4—416no—
158C2×C22⋊C4—44no—
159C2×C42—24no—
160C2×C42—44no—
161C2×C42C2×D414yes—
162C2×C42C2×D412yes—
163C2×C42#4118yes—
164C2×C42C2414yes—
165C2×C42C2412yes—
166C4⋊C8—28no—
167#102—216no—
168#102—48no—
169C22⋊C8—28no—
170#97—48no—
171C4×C8—28no—
172#95—24no—
173#95—24no—
174#95—28no—
175#95—28no—
176#95C2×D418yes—
177C24—24no—
178C24—22no—
179C24—42no—
180C24—44no—
181C24—42no—
182C24—44no—
183C24C22×D411yescharacteristic
184C2×Q8—48no—
185C2×Q8—48no—
186C2×Q8—48no—
187C2×Q8—816no—
188C2×D4—24no—
189C2×D4—24no—
190C2×D4—22no—
191C2×D4—24no—
192C2×D4—416no—
193C2×D4—42no—
194C2×D4—42no—
195C2×D42−1+414yes—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—28no—
199C22×C4—24no—
200C22×C4—24no—
201C22×C4—24no—
202C22×C4—22no—
203C22×C4—44no—
204C22×C4—42no—
205C22×C4—44no—
206C22×C4—42no—
207C22×C4—44no—
208C22×C4—48no—
209C22×C4—44no—
210C22×C4—48no—
211C22×C4—48no—
212C22×C4—48no—
213C22×C4—816no—
214C22×C4#12414yes—
215C22×C4#12414yes—
216C22×C4AGL(1,ℤ/8)18yes—
217C22×C4C22×D418yes—
218C22×C4C22×D411yescharacteristic
219C22×C4C22×D412yes—
220C22×C4C22×D412yes—
221C22×C4C2×#4114yes—
222C22×C4C2511yescharacteristic
223C2×C8—28no—
224C2×C8—28no—
225C2×C8—28no—
226C4⋊C4—28no—
227C4⋊C4—28no—
228C4⋊C4—24no—
229C4⋊C4—28no—
230C4⋊C4—28no—
231C4⋊C4—28no—
232C4⋊C4—432no—
233C4⋊C4—42no—
234C4⋊C4—44no—
235C4⋊C4—48no—
236C4⋊C4—44no—
237C4⋊C4—44no—
238C4⋊C4—82no—
239C4⋊C4—82no—
240C4⋊C4—88no—
241C4⋊C4—88no—
242C22⋊C4—432no—
243C22⋊C4—48no—
244C22⋊C4—48no—
245C42—28no—
246C42—44no—
247C42—48no—
248C42#12418yes—
249C42C2×#4114yes—
250C42C2×#4114yes—
251C23—22no—
252C23—21no—
253C23—21no—
254C23—24no—
255C23—48no—
256C23—416no—
257C23—44no—
258C23—42no—
259C23—42no—
260C23—44no—
261C23—88no—
262C23—84no—
263C23—84no—
264C23C2×#12412yes—
265C23C2×AGL(1,ℤ/8)14yes—
266C23C23×D411yescharacteristic
267Q8—816no—
268D4—28no—
269D4—48no—
270D4—44no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—28no—
276C2×C4—28no—
277C2×C4—28no—
278C2×C4—28no—
279C2×C4—44no—
280C2×C4—416no—
281C2×C4—44no—
282C2×C4—44no—
283C2×C4—42no—
284C2×C4—44no—
285C2×C4—44no—
286C2×C4—44no—
287C2×C4—44no—
288C2×C4—44no—
289C2×C4—48no—
290C2×C4—44no—
291C2×C4—84no—
292C2×C4—88no—
293C2×C4—816no—
294C2×C4#490116yes—
295C2×C4C2×#12412yes—
296C2×C4C2×#12414yes—
297C2×C4C2×#12414yes—
298C2×C4C2×AGL(1,ℤ/8)14yes—
299C2×C4C2×AGL(1,ℤ/8)14yes—
300C2×C4C22×#4112yes—
301C2×C4C2×2−1+412yes—
302C2×C4#58512yes—
303C8—48no—
304C22—24no—
305C22—22no—
306C22—24no—
307C22—22no—
308C22—44no—
309C22—48no—
310C22—816no—
311C22—84no—
312C22—82no—
313C22—82no—
314C22C2×#49014yes—
315C22C22×#12411yescharacteristic
316C22#348511yescharacteristic
317C22#348611yescharacteristic
318C22C22×AGL(1,ℤ/8)12yes—
319C22#358612yes—
320C4—22no—
321C4—24no—
322C4—48no—
323C4—84no—
324C4—84no—
325C4—82no—
326C4—84no—
327C4C2×#49018yes—
328C4#349514yes—
329C2—24no—
330C2—84no—
331C2#3564411yescharacteristic
332C2C22×#49011yescharacteristic
333C2#6062011yescharacteristic
334C2#6171712yes—
335C2#6182712yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35181C2semidirect
4. #27966C2semidirect
5. #27822C2semidirect
6. #27736C2semidirect
7. #21833C2semidirect
9. #20536C2semidirect
16. #2571C22semidirect
18. #2552C22semidirect
27. #2100C22semidirect
30. #1992C22semidirect
31. #1990C22semidirect
32. #1990C22semidirect
33. #1988C22semidirect
35. #1836C22semidirect
41. #1770C22semidirect

Maximal subgroups

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

subgroupindex
#351812
C2×#25422
#291102
#279662
#278222
#277362
#218332
#207322
#205362