Finite Group Databasebeta

← #204633

Subgroups — #204633

Order 512.

1296 conjugacy classes of subgroups; only the 298 normal subgroups are shown.

normal subgroups only

This group has 1296 conjugacy classes of subgroups — too many to draw. Only its 298 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35011C211yesmaximal, characteristic
2#20526C211yesmaximal, characteristic
3C2×#1520C212yesmaximal
4C22×#328C212yesmaximal
5#13966C211yesmaximal, characteristic
6#9683C218yesmaximal, complemented
7C22×C22⋊Q8C2212yes—
8C22×C22⋊C8C2212yes—
9#2857C2214yes—
10#2659—22no—
11#2583—24no—
12#2502C414yescomplemented
13#2434C414yescomplemented
14#2315—22no—
15C2×#380C2211yescharacteristic
16C2×C4×C22⋊C4C2211yescharacteristic
17C22×#95C2211yescharacteristic
18#1752—24no—
19#1749C2214yes—
20#1520—28no—
21C2×#328—28no—
22C2×#328C2218yes—
23C2×#328C2218yes—
24#1442—22no—
25#1434C2214yes—
26C22⋊#95—22no—
27C23×Q8C2×C412yes—
28C24×C4—21no—
29C24×C4C2311yescharacteristic
30C23×C8—22no—
31C2×C4⋊Q8C2×C414yescomplemented
32C2×C22⋊Q8—22no—
33C2×C22⋊Q8—22no—
34C2×C22⋊Q8—24no—
35C2×C22⋊Q8—24no—
36C2×C22⋊Q8—24no—
37C2×C22⋊Q8—24no—
38C2×C22⋊Q8C2×C414yes—
39C2×C22⋊Q8C2314yes—
40C22×C4⋊C4—22no—
41C22×C4⋊C4—21no—
42C22×C4⋊C4C2×C412yes—
43C22×C22⋊C4—22no—
44C22×C22⋊C4—21no—
45C22×C22⋊C4D412yes—
46C22×C42C2312yes—
47C2×C22⋊C8—22no—
48C2×C22⋊C8—24no—
49C2×C22⋊C8D418yes—
50C2×C22⋊C8C2314yes—
51C2×#97—24no—
52#398—28no—
53#395—28no—
54#391C2×C418yescomplemented
55#389—28no—
56#385—28no—
57#385—28no—
58#384C2×C414yescomplemented
59#380—28no—
60#380—24no—
61C4×C22⋊C4C2×C414yes—
62C4×C22⋊C4D414yes—
63C2×#95—24no—
64C2×#95—24no—
65C2×#95—21no—
66C2×#95—24no—
67C2×#95D412yes—
68C2×#95C2312yes—
69C2×#95C2312yes—
70#336—24no—
71#328—216no—
72#328—216no—
73C25C2×D411yescharacteristic
74C22×Q8—22no—
75C22×Q8—24no—
76C22×Q8—24no—
77C22×Q8—84no—
78C22×Q8C22×C414yes—
79C23×C4—22no—
80C23×C4—22no—
81C23×C4—22no—
82C23×C4—22no—
83C23×C4—44no—
84C23×C4—42no—
85C23×C4—44no—
86C23×C4—82no—
87C23×C4—82no—
88C23×C4C22×C412yes—
89C23×C4C2×D414yes—
90C23×C4C2×D411yescharacteristic
91C23×C4C2×D412yes—
92C23×C4C2×D414yes—
93C23×C4#4114yes—
94C23×C4C2411yescharacteristic
95C22×C8—24no—
96C22×C8—28no—
97C22×C8—84no—
98C4⋊Q8—84no—
99C22⋊Q8—28no—
100C22⋊Q8—28no—
101C22⋊Q8—24no—
102C22⋊Q8—24no—
103C22⋊Q8—24no—
104C22⋊Q8—24no—
105C22⋊Q8—44no—
106C22⋊Q8—44no—
107C22⋊Q8—44no—
108C22⋊Q8—44no—
109C2×C4⋊C4—24no—
110C2×C4⋊C4—24no—
111C2×C4⋊C4—24no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—24no—
114C2×C4⋊C4—28no—
115C2×C4⋊C4—28no—
116C2×C4⋊C4—44no—
117C2×C4⋊C4—44no—
118C2×C4⋊C4—42no—
119C2×C4⋊C4—82no—
120C2×C4⋊C4C22×C414yes—
121C2×C22⋊C4—24no—
122C2×C22⋊C4—24no—
123C2×C22⋊C4—24no—
124C2×C22⋊C4—24no—
125C2×C22⋊C4—44no—
126C2×C22⋊C4—44no—
127C2×C22⋊C4—42no—
128C2×C22⋊C4—44no—
129C2×C22⋊C4C22⋊C418yes—
130C2×C42—82no—
131C2×C42C22×C414yes—
132C2×C42C2×D414yes—
133C22⋊C8—28no—
134C22⋊C8—24no—
135C22⋊C8C2×D418yes—
136#97—84no—
137#95—216no—
138#95—24no—
139#95—24no—
140#95—48no—
141#95—42no—
142#95—84no—
143#95C22⋊C418yes—
144C24—42no—
145C24—42no—
146C24C2×C22⋊C412yes—
147C24#12412yes—
148C24#12712yes—
149C24#13714yes—
150C24#13714yes—
151C24C22×D411yescharacteristic
152C2×Q8—24no—
153C2×Q8—24no—
154C2×Q8—48no—
155C2×Q8—44no—
156C2×Q8—84no—
157C2×Q8—88no—
158C2×Q8—88no—
159C22×C4—24no—
160C22×C4—24no—
161C22×C4—24no—
162C22×C4—24no—
163C22×C4—24no—
164C22×C4—42no—
165C22×C4—42no—
166C22×C4—44no—
167C22×C4—44no—
168C22×C4—44no—
169C22×C4—44no—
170C22×C4—44no—
171C22×C4—42no—
172C22×C4—48no—
173C22×C4—42no—
174C22×C4—44no—
175C22×C4—44no—
176C22×C4—44no—
177C22×C4—44no—
178C22×C4—44no—
179C22×C4—42no—
180C22×C4—44no—
181C22×C4—84no—
182C22×C4—84no—
183C22×C4—84no—
184C22×C4—84no—
185C22×C4—816no—
186C22×C4—84no—
187C22×C4—84no—
188C22×C4C2×C22⋊C418yes—
189C22×C4C2×C22⋊C412yes—
190C22×C4C4×D414yes—
191C22×C4C4×D414yes—
192C22×C4#12114yes—
193C22×C4#12114yes—
194C22×C4#12412yes—
195C22×C4#12712yes—
196C22×C4C23×C411yescharacteristic
197C22×C4C22×D411yescharacteristic
198C22×C4C22×D412yes—
199C22×C4C2×#4112yes—
200C2×C8—28no—
201C2×C8—48no—
202C2×C8—88no—
203C4⋊C4—24no—
204C4⋊C4—24no—
205C4⋊C4—44no—
206C4⋊C4—44no—
207C4⋊C4—44no—
208C4⋊C4—48no—
209C4⋊C4—88no—
210C4⋊C4—88no—
211C4⋊C4—84no—
212C22⋊C4—216no—
213C22⋊C4—44no—
214C22⋊C4—44no—
215C22⋊C4—48no—
216C22⋊C4—84no—
217C42—44no—
218C42—44no—
219C23—22no—
220C23—24no—
221C23—22no—
222C23—24no—
223C23—42no—
224C23—42no—
225C23—42no—
226C23—48no—
227C23—42no—
228C23—48no—
229C23#39014yes—
230C23#41918yes—
231C23#49118yes—
232C23#49714yes—
233C23#49714yes—
234C23C22×C22⋊C411yescharacteristic
235C23C2×#12411yescharacteristic
236C23C2×#12711yescharacteristic
237C23C2×#13712yes—
238C23C2×#13712yes—
239Q8—84no—
240Q8—84no—
241Q8—88no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—24no—
245C2×C4—28no—
246C2×C4—28no—
247C2×C4—28no—
248C2×C4—42no—
249C2×C4—44no—
250C2×C4—44no—
251C2×C4—44no—
252C2×C4—44no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—44no—
257C2×C4—48no—
258C2×C4—48no—
259C2×C4—48no—
260C2×C4—44no—
261C2×C4—44no—
262C2×C4—84no—
263C2×C4—84no—
264C2×C4—84no—
265C2×C4—82no—
266C2×C4—88no—
267C2×C4—88no—
268C2×C4—816no—
269C2×C4—82no—
270C2×C4—82no—
271C2×C4—82no—
272C2×C4—88no—
273C2×C4—82no—
274C2×C4#39014yes—
275C2×C4C2×C4×D412yes—
276C2×C4C2×#12112yes—
277C8—48no—
278C22—21no—
279C22—24no—
280C22—21no—
281C22—24no—
282C22—24no—
283C22—216no—
284C22—21no—
285C22—24no—
286C22—21no—
287C22—44no—
288C22—48no—
289C22—44no—
290C22#196018yes—
291C22C2×#39011yescharacteristic
292C22#288312yes—
293C22C2×#41912yes—
294C22#290112yes—
295C22C2×#49112yes—
296C22C2×#49711yescharacteristic
297C22C2×#49711yescharacteristic
298C4—22no—
299C4—22no—
300C4—24no—
301C4—48no—
302C4—84no—
303C4—84no—
304C4—84no—
305C4—162no—
306C2—21no—
307C2—21no—
308C2—22no—
309C2—28no—
310C2#1762114yes—
311C2C2×#196011yescharacteristic
312C2#2121311yescharacteristic
313C2#2125311yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #9683C2semidirect
12. #2502C4semidirect
13. #2434C4semidirect
31. C2×C4⋊Q8C2×C4semidirect
54. #391C2×C4semidirect
58. #384C2×C4semidirect

Maximal subgroups

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

subgroupindex
#350112
#205262
C2×#15202
C22×#3282
#139662
#96832