Finite Group Databasebeta

← #6510736

Subgroups — #6510736

Order 512.

1660 conjugacy classes of subgroups.

The subgroup lattice has 301 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35020C211yesmaximal, characteristic
2C2×#2502C211yesmaximal, characteristic
3#29112C214yesmaximal, complemented
4#27965C214yesmaximal, complemented
5#27715C2116yesmaximal, complemented
6#21759C212yesmaximal
7C2×#1824C212yesmaximal
8#20526C211yesmaximal, characteristic
9C22×C22⋊Q8C2211yescharacteristic
10C23×C4⋊C4C2211yescharacteristic
11C22×C22⋊C8C2212yes—
12#2851C2214yes—
13#2646—22no—
14#2569—24no—
15#2502—24no—
16#2502C2218yes—
17#2442C2218yes—
18C2×#398—22no—
19#2315—22no—
20C2×#391C2212yes—
21C2×#385—22no—
22C2×#384C2212yes—
23C2×#380C2212yes—
24C2×C4×C22⋊C4C2211yescharacteristic
25#2099C22116yescomplemented
26#2074C2218yescomplemented
27#2026C2218yescomplemented
28#1992C2218yescomplemented
29#1935C22116yescomplemented
30#1892C2218yes—
31#1872C2218yes—
32#1836C2218yescomplemented
33#1824C2214yes—
34#1824C2218yes—
35#1824C2218yes—
36#1787C2214yes—
37#1752C2218yes—
38#1749C2214yes—
39C2×#340C2218yes—
40C23×Q8C2311yescharacteristic
41C24×C4—21no—
42C24×C4C2311yescharacteristic
43C23×C8C2314yes—
44C2×#125C2314yes—
45C2×C22⋊Q8—22no—
46C2×C22⋊Q8—28no—
47C2×C22⋊Q8C2314yes—
48C22×C4⋊C4—28no—
49C22×C4⋊C4—21no—
50C22×C4⋊C4—21no—
51C22×C4⋊C4—22no—
52C22×C4⋊C4—24no—
53C22×C4⋊C4C2312yes—
54C22×C4⋊C4C2311yescharacteristic
55C22×C4⋊C4C2314yes—
56C22×C4⋊C4C2314yes—
57C22×C22⋊C4—22no—
58C22×C22⋊C4—21no—
59C22×C22⋊C4D412yes—
60C22×C42C2312yes—
61C2×#103C23116yes—
62C2×C22⋊C8—28no—
63C2×C22⋊C8C2314yes—
64C2×C22⋊C8C2314yes—
65C2×#97C2318yes—
66#400—28no—
67#398—28no—
68#398—216no—
69#391—24no—
70#391C2318yes—
71#389C2318yes—
72#387—28no—
73#385—28no—
74#385—28no—
75#385—28no—
76#385—28no—
77#384—24no—
78#384C2318yes—
79#382C2314yes—
80#380—22no—
81#380—22no—
82#380—28no—
83#380C2314yes—
84#380C2314yes—
85C4×C22⋊C4D414yes—
86C4×C22⋊C4C2314yes—
87C2×#95—22no—
88C2×#95—24no—
89C2×#95D412yes—
90#340C23116yes—
91#340C23116yes—
92#340C23116yes—
93#336C2318yes—
94C25C2×D411yescharacteristic
95C22×Q8—21no—
96C22×Q8—24no—
97C22×Q8—44no—
98C22×Q8C2412yes—
99C23×C4—21no—
100C23×C4—21no—
101C23×C4—22no—
102C23×C4—24no—
103C23×C4—24no—
104C23×C4—44no—
105C23×C4—44no—
106C23×C4—84no—
107C23×C4C2×D414yes—
108C23×C4C2×D411yescharacteristic
109C23×C4C2×D412yes—
110C23×C4#4118yes—
111C23×C4C2412yes—
112C23×C4C2411yescharacteristic
113C22×C8—28no—
114C22×C8—48no—
115C22×C8C2418yes—
116#125—48no—
117C22⋊Q8—28no—
118C22⋊Q8—28no—
119C22⋊Q8—48no—
120C2×C4⋊C4—28no—
121C2×C4⋊C4—28no—
122C2×C4⋊C4—28no—
123C2×C4⋊C4—24no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—24no—
126C2×C4⋊C4—24no—
127C2×C4⋊C4—21no—
128C2×C4⋊C4—216no—
129C2×C4⋊C4—24no—
130C2×C4⋊C4—28no—
131C2×C4⋊C4—28no—
132C2×C4⋊C4—48no—
133C2×C4⋊C4—44no—
134C2×C4⋊C4—48no—
135C2×C4⋊C4—44no—
136C2×C4⋊C4—88no—
137C2×C4⋊C4C2414yes—
138C2×C4⋊C4C2412yes—
139C2×C4⋊C4C2418yes—
140C2×C22⋊C4—28no—
141C2×C22⋊C4—24no—
142C2×C22⋊C4—24no—
143C2×C22⋊C4—24no—
144C2×C22⋊C4—44no—
145C2×C22⋊C4—48no—
146C2×C22⋊C4C2×D414yes—
147C2×C22⋊C4C2×D414yes—
148C2×C42—44no—
149C2×C42C2×D414yes—
150C2×C42C2414yes—
151#103—416no—
152C22⋊C8—28no—
153C22⋊C8—28no—
154#97—48no—
155#95—24no—
156#95—24no—
157#95—216no—
158#95—28no—
159#95—28no—
160#95C2×D418yes—
161C24—42no—
162C24—42no—
163C24#12414yes—
164C24#13718yes—
165C24C22×D412yes—
166C24C22×D411yescharacteristic
167C2×Q8—24no—
168C2×Q8—28no—
169C2×Q8—44no—
170C2×Q8—42no—
171C2×Q8—88no—
172C22×C4—24no—
173C22×C4—28no—
174C22×C4—28no—
175C22×C4—22no—
176C22×C4—28no—
177C22×C4—24no—
178C22×C4—22no—
179C22×C4—416no—
180C22×C4—44no—
181C22×C4—44no—
182C22×C4—44no—
183C22×C4—42no—
184C22×C4—44no—
185C22×C4—48no—
186C22×C4—44no—
187C22×C4—42no—
188C22×C4—42no—
189C22×C4—44no—
190C22×C4—48no—
191C22×C4—48no—
192C22×C4—88no—
193C22×C4—84no—
194C22×C4—84no—
195C22×C4—816no—
196C22×C4#12414yes—
197C22×C4#12418yes—
198C22×C4C22×D418yes—
199C22×C4C22×D411yescharacteristic
200C22×C4C22×D412yes—
201C22×C4C2×#4114yes—
202C22×C4C2×#4114yes—
203C22×C4C2×#4114yes—
204C22×C4C2511yescharacteristic
205C2×C8—28no—
206C2×C8—216no—
207C2×C8—88no—
208C4⋊C4—28no—
209C4⋊C4—24no—
210C4⋊C4—216no—
211C4⋊C4—48no—
212C4⋊C4—44no—
213C4⋊C4—42no—
214C4⋊C4—48no—
215C4⋊C4—88no—
216C4⋊C4—84no—
217C4⋊C4—816no—
218C4⋊C4—88no—
219C22⋊C4—28no—
220C22⋊C4—28no—
221C22⋊C4—48no—
222C22⋊C4—48no—
223C22⋊C4—48no—
224C42—28no—
225C42—44no—
226C23—22no—
227C23—24no—
228C23—22no—
229C23—24no—
230C23—44no—
231C23—44no—
232C23—416no—
233C23#491116yes—
234C23C2×#12414yes—
235C23C2×#12412yes—
236C23C2×#13714yes—
237C23C2×#13714yes—
238C23C2×#13714yes—
239C23C23×D411yescharacteristic
240Q8—48no—
241Q8—84no—
242C2×C4—22no—
243C2×C4—24no—
244C2×C4—28no—
245C2×C4—28no—
246C2×C4—28no—
247C2×C4—28no—
248C2×C4—28no—
249C2×C4—28no—
250C2×C4—44no—
251C2×C4—48no—
252C2×C4—416no—
253C2×C4—44no—
254C2×C4—48no—
255C2×C4—48no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—44no—
259C2×C4—48no—
260C2×C4—44no—
261C2×C4—48no—
262C2×C4—84no—
263C2×C4—88no—
264C2×C4—84no—
265C2×C4—816no—
266C2×C4—168no—
267C2×C4C2×#12414yes—
268C2×C4C2×#12412yes—
269C2×C4C22×#4112yes—
270C2×C4#58514yes—
271C8—48no—
272C22—28no—
273C22—28no—
274C22—28no—
275C22—22no—
276C22—24no—
277C22—22no—
278C22—24no—
279C22—416no—
280C22C2×#49118yes—
281C22C2×#49114yes—
282C22C22×#12411yescharacteristic
283C22#348512yes—
284C22C22×#13712yes—
285C22#358512yes—
286C4—24no—
287C4—22no—
288C4—48no—
289C4—88no—
290C4—82no—
291C4—82no—
292C4—84no—
293C4#349114yes—
294C2—24no—
295C2—28no—
296C2#3545511yescharacteristic
297C2C22×#49111yescharacteristic
298C2#6061911yescharacteristic
299C2#6170914yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #29112C2semidirect
4. #27965C2semidirect
5. #27715C2semidirect
25. #2099C22semidirect
26. #2074C22semidirect
27. #2026C22semidirect
28. #1992C22semidirect
29. #1935C22semidirect
32. #1836C22semidirect

Maximal subgroups

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

subgroupindex
#350202
C2×#25022
#291122
#279652
#277152
#217592
C2×#18242
#205262