Finite Group Databasebeta

← #6528492

Subgroups — #6528492

Order 512.

1652 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35181C211yesmaximal, characteristic, complemented
2C2×#2563C211yesmaximal, characteristic
3#29104C214yesmaximal
4#27964C214yesmaximal, complemented
5#27814C218yesmaximal, complemented
6#27728C218yesmaximal, complemented
7#21834C212yesmaximal, complemented
8#20732C212yesmaximal
9#20533C211yesmaximal, characteristic, complemented
10C22×C4⋊Q8C2211yescharacteristic
11C22×C4×D4C2211yescharacteristic
12C2×#431C2212yes—
13#2851C2214yes—
14#2646—22no—
15#2571C2214yescomplemented
16#2570—24no—
17#2563—24no—
18#2563C2218yes—
19#2552C2214yescomplemented
20C2×#393—22no—
21#2317—22no—
22C2×#390C2212yes—
23#2285C2212yes—
24C2×#382C2212yes—
25C2×#380—22no—
26C2×C4×C4⋊C4C2211yescharacteristic
27#2101C22116yescomplemented
28#2073C2218yes—
29#2025C2218yes—
30#1991C2218yescomplemented
31#1990C2214yescomplemented
32#1990C2214yescomplemented
33#1987C2218yescomplemented
34#1934C22116yes—
35#1835C2218yescomplemented
36#1824C2218yes—
37#1823C2214yes—
38#1802C2214yes—
39#1802C2214yes—
40#1786C2214yes—
41C8×C4⋊C4C2218yescomplemented
42C4×#102C2218yes—
43C4×C22⋊C8C2214yes—
44C23×Q8—21no—
45C23×D4C2311yescharacteristic
46C24×C4—21no—
47C2×C4⋊Q8—22no—
48C2×C4⋊Q8—28no—
49C2×C4⋊Q8C2314yes—
50C2×#125C2314yes—
51C2×C4×D4—22no—
52C2×C4×D4—28no—
53C2×C4×D4C2314yes—
54C2×#117—22no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4D412yes—
57C22×C4⋊C4C2311yescharacteristic
58C22×C4⋊C4C2312yes—
59C22×C22⋊C4—21no—
60C22×C22⋊C4—24no—
61C22×C42C2312yes—
62C22×C42C2311yescharacteristic
63#431—28no—
64C2×C4⋊C8C2314yes—
65C2×#102C2318yes—
66C2×#102C2318yes—
67C2×C22⋊C8C2314yes—
68C2×C4×C8C2314yes—
69C2×C4×C8C2318yes—
70#400—28no—
71#398—28no—
72#393—216no—
73#390—24no—
74#390C2318yes—
75#389—24no—
76#389—24no—
77#389C2318yes—
78#387—28no—
79#385—28no—
80#384—24no—
81#384C2312yes—
82#384C2312yes—
83#382—24no—
84#382—24no—
85#382—22no—
86#382C2318yes—
87#382C2314yes—
88#380—22no—
89#380—28no—
90#380—28no—
91#380C2314yes—
92C4×C4⋊C4—22no—
93C4×C4⋊C4D414yes—
94C4×C4⋊C4C2314yes—
95C4×C4⋊C4C2314yes—
96C2×#95—24no—
97C2×#95D412yes—
98#340C2318yes—
99#340C2318yes—
100#340C23116yes—
101#340C23116yes—
102#336C2318yes—
103C25—21no—
104C22×Q8—24no—
105C22×Q8—44no—
106C22×Q8—84no—
107C22×D4—24no—
108C22×D4—21no—
109C22×D4—24no—
110C22×D4—42no—
111C22×D4C2412yes—
112C23×C4—22no—
113C23×C4—21no—
114C23×C4—24no—
115C23×C4—44no—
116C23×C4—48no—
117C23×C4C2×D414yes—
118C23×C4C2×D411yescharacteristic
119C23×C4C2×D411yescharacteristic
120C23×C4C2411yescharacteristic
121C22×C8C2418yes—
122C4⋊Q8—28no—
123C4⋊Q8—28no—
124C4⋊Q8—48no—
125#125—48no—
126C4×D4—28no—
127C4×D4—28no—
128C4×D4—48no—
129#117—48no—
130C2×C4⋊C4—28no—
131C2×C4⋊C4—24no—
132C2×C4⋊C4—28no—
133C2×C4⋊C4—24no—
134C2×C4⋊C4—21no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—28no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—28no—
142C2×C4⋊C4—44no—
143C2×C4⋊C4C2×D414yes—
144C2×C4⋊C4C2×D414yes—
145C2×C4⋊C4C2412yes—
146C2×C4⋊C4C2414yes—
147C2×C4⋊C4C2414yes—
148C2×C4⋊C4C2414yes—
149C2×C22⋊C4—24no—
150C2×C22⋊C4—24no—
151C2×C22⋊C4—216no—
152C2×C22⋊C4—44no—
153C2×C22⋊C4—88no—
154C2×C42—24no—
155C2×C42—44no—
156C2×C42C2×D414yes—
157C2×C42C2×D412yes—
158C2×C42#4118yes—
159C2×C42C2414yes—
160C2×C42C2412yes—
161C4⋊C8—28no—
162#102—216no—
163#102—48no—
164C22⋊C8—28no—
165C4×C8—28no—
166C4×C8—48no—
167#95—216no—
168#95—24no—
169#95—24no—
170#95—28no—
171#95—28no—
172#95C2×D418yes—
173C24—24no—
174C24—22no—
175C24—44no—
176C24—42no—
177C24—42no—
178C24—44no—
179C24C22×D411yescharacteristic
180C2×Q8—48no—
181C2×Q8—84no—
182C2×Q8—84no—
183C2×Q8—816no—
184C2×D4—24no—
185C2×D4—24no—
186C2×D4—22no—
187C2×D4—24no—
188C2×D4—416no—
189C2×D4—42no—
190C2×D4—42no—
191C2×D42−1+414yes—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—28no—
195C22×C4—24no—
196C22×C4—22no—
197C22×C4—24no—
198C22×C4—24no—
199C22×C4—42no—
200C22×C4—44no—
201C22×C4—44no—
202C22×C4—48no—
203C22×C4—42no—
204C22×C4—44no—
205C22×C4—44no—
206C22×C4—48no—
207C22×C4—48no—
208C22×C4—48no—
209C22×C4—816no—
210C22×C4#12414yes—
211C22×C4#12414yes—
212C22×C4#13518yes—
213C22×C4C22×D418yes—
214C22×C4C22×D411yescharacteristic
215C22×C4C22×D412yes—
216C22×C4C22×D412yes—
217C22×C4C2×#4114yes—
218C22×C4C2511yescharacteristic
219C2×C8—28no—
220C2×C8—28no—
221C2×C8—28no—
222C4⋊C4—28no—
223C4⋊C4—28no—
224C4⋊C4—24no—
225C4⋊C4—28no—
226C4⋊C4—28no—
227C4⋊C4—28no—
228C4⋊C4—42no—
229C4⋊C4—44no—
230C4⋊C4—48no—
231C4⋊C4—44no—
232C4⋊C4—44no—
233C4⋊C4—82no—
234C4⋊C4—82no—
235C4⋊C4—88no—
236C4⋊C4—88no—
237C22⋊C4—48no—
238C22⋊C4—48no—
239C22⋊C4—816no—
240C42—28no—
241C42—44no—
242C42—48no—
243C42#12418yes—
244C42C2×#4114yes—
245C42C2×#4114yes—
246C23—22no—
247C23—21no—
248C23—21no—
249C23—24no—
250C23—48no—
251C23—416no—
252C23—44no—
253C23—42no—
254C23—42no—
255C23—44no—
256C23—88no—
257C23—84no—
258C23—84no—
259C23C2×#12412yes—
260C23C2×#13514yes—
261C23C23×D411yescharacteristic
262Q8—816no—
263D4—28no—
264D4—48no—
265D4—44no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—28no—
271C2×C4—28no—
272C2×C4—28no—
273C2×C4—28no—
274C2×C4—44no—
275C2×C4—48no—
276C2×C4—44no—
277C2×C4—44no—
278C2×C4—42no—
279C2×C4—44no—
280C2×C4—44no—
281C2×C4—44no—
282C2×C4—44no—
283C2×C4—44no—
284C2×C4—44no—
285C2×C4—88no—
286C2×C4—88no—
287C2×C4—84no—
288C2×C4—816no—
289C2×C4#488116yes—
290C2×C4C2×#12412yes—
291C2×C4C2×#12414yes—
292C2×C4C2×#12414yes—
293C2×C4C2×#13514yes—
294C2×C4C2×#13514yes—
295C2×C4C22×#4112yes—
296C2×C4C2×2−1+412yes—
297C2×C4#58512yes—
298C8—48no—
299C22—24no—
300C22—22no—
301C22—24no—
302C22—22no—
303C22—44no—
304C22—48no—
305C22—816no—
306C22—84no—
307C22—82no—
308C22—82no—
309C22C2×#48814yes—
310C22C22×#12411yescharacteristic
311C22#348511yescharacteristic
312C22#348611yescharacteristic
313C22C22×#13512yes—
314C22#358512yes—
315C4—22no—
316C4—24no—
317C4—48no—
318C4—84no—
319C4—84no—
320C4—82no—
321C4—84no—
322C4C2×#48818yes—
323C4#349514yes—
324C2—24no—
325C2—84no—
326C2#3564411yescharacteristic
327C2C22×#48811yescharacteristic
328C2#6061911yescharacteristic
329C2#6170012yes—
330C2#6181912yes—
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. #27964C2semidirect
5. #27814C2semidirect
6. #27728C2semidirect
7. #21834C2semidirect
9. #20533C2semidirect
15. #2571C22semidirect
19. #2552C22semidirect
27. #2101C22semidirect
30. #1991C22semidirect
31. #1990C22semidirect
32. #1990C22semidirect
33. #1987C22semidirect
35. #1835C22semidirect
41. C8×C4⋊C4C22semidirect

Maximal subgroups

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

subgroupindex
#351812
C2×#25632
#291042
#279642
#278142
#277282
#218342
#207322
#205332