Finite Group Databasebeta

← #207017

Subgroups — #207017

Order 512.

858 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35181C211yesmaximal, characteristic
2#20534C211yesmaximal, characteristic
3#13964C211yesmaximal, characteristic
4#9887C214yesmaximal
5#9886C214yesmaximal
6#8022C212yesmaximal
7#8018C212yesmaximal
8C2×#432C2212yes—
9#2851C2212yes—
10#2851C2212yes—
11#2646—22no—
12#2571C414yescomplemented
13#2570—24no—
14#2552C414yescomplemented
15#2317—22no—
16#2285C2212yes—
17C2×#383C2211yescharacteristic
18C2×C4×C4⋊C4C2211yescharacteristic
19C2×#376C2211yescharacteristic
20#1771—24no—
21#1749C2212yes—
22C4×C22⋊C8C2212yes—
23#1441—22no—
24#1436—22no—
25C4⋊#95C2212yes—
26#1434C2212yes—
27#1294C2218yes—
28#1294C2218yes—
29C2×#125C2×C412yes—
30C2×#125C2×C412yes—
31C2×#117—22no—
32C22×C4⋊C4—22no—
33C22×C4⋊C4D412yes—
34C22×C42—21no—
35C22×C42C2311yescharacteristic
36C22×C42C2311yescharacteristic
37C22×C42C2311yescharacteristic
38#432—24no—
39#432Q818yes—
40C2×C4⋊C8—22no—
41C2×C22⋊C8C2314yes—
42C2×#97—22no—
43C2×#97—22no—
44C2×C4×C8—22no—
45#400—24no—
46#400—24no—
47#398—24no—
48#398—24no—
49#389—24no—
50#389—24no—
51#389C2×C414yes—
52#389C2×C414yes—
53#387—24no—
54#387—24no—
55#385—24no—
56#385—24no—
57#384—24no—
58#384C2×C412yes—
59#384C2×C412yes—
60#383—24no—
61#382—24no—
62#382—22no—
63#382C2×C412yes—
64#382C2×C412yes—
65#380—22no—
66#380C2314yes—
67C4×C4⋊C4—22no—
68C4×C4⋊C4C2×C414yes—
69C4×C4⋊C4C2×C414yes—
70C4×C4⋊C4D414yes—
71#376—24no—
72C2×#95—22no—
73C2×#95—22no—
74C2×#95—21no—
75C2×#95—21no—
76C2×#95—21no—
77C2×#95—22no—
78C2×#95D412yes—
79C2×#95C2312yes—
80C2×#95C2312yes—
81#336—22no—
82#336—22no—
83C23×C4—22no—
84C23×C4—22no—
85C23×C4C2×D411yescharacteristic
86C23×C4C2×D412yes—
87C23×C4C2×D412yes—
88C23×C4C2×D411yescharacteristic
89C23×C4C2411yescharacteristic
90C22×C8—24no—
91#125—82no—
92#125—82no—
93#117—44no—
94#117—44no—
95C2×C4⋊C4—22no—
96C2×C4⋊C4—22no—
97C2×C4⋊C4—24no—
98C2×C4⋊C4—24no—
99C2×C4⋊C4—22no—
100C2×C4⋊C4—22no—
101C2×C4⋊C4—24no—
102C2×C4⋊C4—24no—
103C2×C4⋊C4—42no—
104C2×C4⋊C4—42no—
105C2×C4⋊C4—44no—
106C2×C4⋊C4—44no—
107C2×C4⋊C4—82no—
108C2×C4⋊C4C22⋊C414yes—
109C2×C4⋊C4C22⋊C414yes—
110C2×C4⋊C4C22×C414yes—
111C2×C4⋊C4C22×C414yes—
112C2×C22⋊C4—24no—
113C2×C42—24no—
114C2×C42—44no—
115C2×C42—42no—
116C2×C42—81no—
117C2×C42—81no—
118C2×C42C22×C412yes—
119C2×C42C22×C412yes—
120C2×C42C22×C412yes—
121C2×C42C2×D412yes—
122C2×C42C2×D412yes—
123C2×C42C2×D412yes—
124C2×C42C2×Q814yes—
125C2×C42#4114yes—
126C4⋊C8—28no—
127C22⋊C8—24no—
128C22⋊C8C2×Q818yes—
129#97—28no—
130#97—82no—
131C4×C8—82no—
132#95—24no—
133#95—24no—
134#95—24no—
135#95—24no—
136#95—24no—
137#95—24no—
138#95—24no—
139#95—24no—
140#95—44no—
141#95—44no—
142#95—42no—
143#95—42no—
144#95—42no—
145#95—82no—
146#95C22⋊C414yes—
147#95C22⋊C414yes—
148C24C22×D411yescharacteristic
149C22×C4—22no—
150C22×C4—22no—
151C22×C4—22no—
152C22×C4—22no—
153C22×C4—22no—
154C22×C4—22no—
155C22×C4—24no—
156C22×C4—24no—
157C22×C4—24no—
158C22×C4—42no—
159C22×C4—41no—
160C22×C4—42no—
161C22×C4—42no—
162C22×C4—41no—
163C22×C4—42no—
164C22×C4—42no—
165C22×C4—42no—
166C22×C4—42no—
167C22×C4—42no—
168C22×C4—42no—
169C22×C4—42no—
170C22×C4—42no—
171C22×C4—42no—
172C22×C4—42no—
173C22×C4—42no—
174C22×C4—42no—
175C22×C4—42no—
176C22×C4—42no—
177C22×C4—42no—
178C22×C4C2×C22⋊C414yes—
179C22×C4C2×C22⋊C414yes—
180C22×C4C2×C22⋊C412yes—
181C22×C4C2×C22⋊C412yes—
182C22×C4#12412yes—
183C22×C4#12412yes—
184C22×C4C4⋊Q812yes—
185C22×C4C4⋊Q812yes—
186C22×C4C23×C411yescharacteristic
187C22×C4C22×D411yescharacteristic
188C22×C4C22×Q812yes—
189C22×C4C2×#4112yes—
190C2×C8—28no—
191C2×C8—44no—
192C2×C8—44no—
193C4⋊C4—44no—
194C4⋊C4—44no—
195C4⋊C4—44no—
196C4⋊C4—44no—
197C4⋊C4—44no—
198C4⋊C4—44no—
199C4⋊C4—82no—
200C4⋊C4—82no—
201C4⋊C4—84no—
202C4⋊C4—84no—
203C4⋊C4—84no—
204C4⋊C4—84no—
205C4⋊C4—88no—
206C22⋊C4—44no—
207C22⋊C4—44no—
208C42—42no—
209C42—42no—
210C42—42no—
211C42—42no—
212C42—48no—
213C42—48no—
214C42C4×Q818yes—
215C42C22⋊Q814yes—
216C42C22⋊Q814yes—
217C23—21no—
218C23—21no—
219C23—22no—
220C23C22×C22⋊C411yescharacteristic
221C23C2×#12411yescharacteristic
222C23C2×C4⋊Q811yescharacteristic
223C23#57512yes—
224C23#57512yes—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—24no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—24no—
237C2×C4—24no—
238C2×C4—24no—
239C2×C4—24no—
240C2×C4—24no—
241C2×C4—24no—
242C2×C4—42no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—44no—
247C2×C4—44no—
248C2×C4—44no—
249C2×C4—44no—
250C2×C4—44no—
251C2×C4—44no—
252C2×C4—44no—
253C2×C4—44no—
254C2×C4—44no—
255C2×C4—44no—
256C2×C4—44no—
257C2×C4—82no—
258C2×C4—82no—
259C2×C4—82no—
260C2×C4—82no—
261C2×C4—82no—
262C2×C4—82no—
263C2×C4—82no—
264C2×C4—82no—
265C2×C4—82no—
266C2×C4—82no—
267C2×C4—82no—
268C2×C4—82no—
269C2×C4—82no—
270C2×C4#39114yes—
271C2×C4#39114yes—
272C2×C4C2×C4×Q812yes—
273C2×C4C2×C22⋊Q812yes—
274C8—48no—
275C22—21no—
276C22—21no—
277C22—21no—
278C22—21no—
279C22—22no—
280C22—22no—
281C22—22no—
282C22—22no—
283C22C2×#39111yescharacteristic
284C22#288311yescharacteristic
285C22#288411yescharacteristic
286C22#289612yes—
287C22#290112yes—
288C22#314212yes—
289C22#316611yescharacteristic
290C22#316611yescharacteristic
291C4—24no—
292C4—24no—
293C4—22no—
294C4—22no—
295C4—44no—
296C4—44no—
297C4—84no—
298C4—84no—
299C4—84no—
300C4—162no—
301C2—22no—
302C2—21no—
303C2—21no—
304C2#1762412yes—
305C2#1762912yes—
306C2#2130711yescharacteristic
307C2#2132511yescharacteristic
308C2#2133411yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
12. #2571C4semidirect
14. #2552C4semidirect

Maximal subgroups

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

subgroupindex
#351812
#205342
#139642
#98872
#98862
#80222
#80182