Finite Group Databasebeta

← #207278

Subgroups — #207278

Order 512.

870 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35181C211yesmaximal, characteristic
2#21955C211yesmaximal, characteristic
3#13806C211yesmaximal, characteristic
4#9913C214yesmaximal
5#9910C214yesmaximal
6#8022C212yesmaximal
7#8018C212yesmaximal
8C2×#432C2212yes—
9#2851C2214yes—
10#2646—21no—
11#2646—21no—
12#2571C414yescomplemented
13#2570—24no—
14#2552C414yescomplemented
15#2317—22no—
16#2285C2212yes—
17C2×#383C2211yescharacteristic
18C2×C4×C4⋊C4C2211yescharacteristic
19C2×#376C2211yescharacteristic
20#1940—24no—
21#1797C2214yes—
22#1436—22no—
23#1434—21no—
24#1434C2212yes—
25C4×#95—21no—
26C4×#95C2212yes—
27#1294C2218yes—
28#1294C2214yes—
29#1294C2214yes—
30C2×#125C2×C414yes—
31C2×#117—22no—
32C22×C4⋊C4D412yes—
33C22×C42—21no—
34C22×C42—22no—
35C22×C42C2312yes—
36C22×C42C2311yescharacteristic
37#432—24no—
38#432—24no—
39C2×C4⋊C8—22no—
40C2×C4⋊C8—24no—
41C2×C22⋊C8C2314yes—
42C2×#97—22no—
43#400—24no—
44#400—24no—
45#398—24no—
46#398—24no—
47#389—24no—
48#389—24no—
49#389C2×C414yes—
50#389C2×C414yes—
51#387—24no—
52#387—24no—
53#385—24no—
54#385—24no—
55#384—24no—
56#384C2×C412yes—
57#384C2×C412yes—
58#383—22no—
59#383Q814yes—
60#382—24no—
61#382—22no—
62#382C2×C414yes—
63#380—22no—
64#380C2314yes—
65C4×C4⋊C4—22no—
66C4×C4⋊C4C2×C414yes—
67C4×C4⋊C4C2×C414yes—
68C4×C4⋊C4D414yes—
69#376—22no—
70#376Q814yes—
71C2×#95—21no—
72C2×#95—21no—
73C2×#95—21no—
74C2×#95—22no—
75C2×#95—22no—
76C2×#95—22no—
77C2×#95D411yescharacteristic
78C2×#95D411yescharacteristic
79C2×#95C2311yescharacteristic
80C2×#95C2312yes—
81C2×#95C2311yescharacteristic
82#336—24no—
83C23×C4—21no—
84C23×C4—22no—
85C23×C4—21no—
86C23×C4C2×D411yescharacteristic
87C23×C4C2×D412yes—
88C23×C4C2×D412yes—
89C23×C4C2×D411yescharacteristic
90C23×C4C2411yescharacteristic
91C22×C8—24no—
92#125—84no—
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⋊C4C22⋊C414yes—
106C2×C4⋊C4C22⋊C414yes—
107C2×C4⋊C4C22×C414yes—
108C2×C4⋊C4C22×C414yes—
109C2×C22⋊C4—24no—
110C2×C42—24no—
111C2×C42—24no—
112C2×C42—44no—
113C2×C42—44no—
114C2×C42—44no—
115C2×C42—82no—
116C2×C42—82no—
117C2×C42C22×C414yes—
118C2×C42C22×C412yes—
119C2×C42C2×D414yes—
120C2×C42C2×D412yes—
121C2×C42C2×Q812yes—
122C2×C42C2×Q812yes—
123C2×C42#4112yes—
124C2×C42#4112yes—
125C4⋊C8—44no—
126C4⋊C8—84no—
127C22⋊C8—24no—
128C22⋊C8—24no—
129#97—44no—
130#95—24no—
131#95—24no—
132#95—24no—
133#95—24no—
134#95—24no—
135#95—24no—
136#95—24no—
137#95—24no—
138#95—24no—
139#95—21no—
140#95—21no—
141#95—22no—
142#95—44no—
143#95—44no—
144#95—82no—
145#95C22⋊C414yes—
146#95C22⋊C414yes—
147#95C2×Q812yes—
148#95C2×Q812yes—
149#95C2×Q814yes—
150C24C22×D411yescharacteristic
151C22×C4—22no—
152C22×C4—22no—
153C22×C4—24no—
154C22×C4—22no—
155C22×C4—22no—
156C22×C4—22no—
157C22×C4—22no—
158C22×C4—22no—
159C22×C4—24no—
160C22×C4—24no—
161C22×C4—24no—
162C22×C4—22no—
163C22×C4—42no—
164C22×C4—42no—
165C22×C4—42no—
166C22×C4—41no—
167C22×C4—41no—
168C22×C4—42no—
169C22×C4—42no—
170C22×C4—41no—
171C22×C4—41no—
172C22×C4—42no—
173C22×C4—42no—
174C22×C4—41no—
175C22×C4—42no—
176C22×C4—42no—
177C22×C4—42no—
178C22×C4—42no—
179C22×C4—41no—
180C22×C4—42no—
181C22×C4—42no—
182C22×C4—42no—
183C22×C4C2×C22⋊C414yes—
184C22×C4C2×C22⋊C414yes—
185C22×C4C2×C22⋊C412yes—
186C22×C4C2×C22⋊C412yes—
187C22×C4C22⋊Q812yes—
188C22×C4C22⋊Q812yes—
189C22×C4C22⋊Q812yes—
190C22×C4C22⋊Q812yes—
191C22×C4C23×C411yescharacteristic
192C22×C4C22×D411yescharacteristic
193C22×C4C22×Q811yescharacteristic
194C22×C4C22×Q811yescharacteristic
195C22×C4C2×#4111yescharacteristic
196C22×C4C2×#4111yescharacteristic
197C2×C8—44no—
198C2×C8—44no—
199C2×C8—44no—
200C4⋊C4—44no—
201C4⋊C4—44no—
202C4⋊C4—44no—
203C4⋊C4—44no—
204C4⋊C4—44no—
205C4⋊C4—44no—
206C4⋊C4—82no—
207C4⋊C4—82no—
208C4⋊C4—84no—
209C4⋊C4—84no—
210C4⋊C4—84no—
211C4⋊C4—84no—
212C22⋊C4—44no—
213C22⋊C4—44no—
214C42—44no—
215C42—44no—
216C42—48no—
217C42—48no—
218C42—88no—
219C42C4×Q814yes—
220C42C4×Q814yes—
221C42#12414yes—
222C42C4⋊Q814yes—
223C23—21no—
224C23—21no—
225C23—22no—
226C23C22×C22⋊C411yescharacteristic
227C23C2×C22⋊Q811yescharacteristic
228C23C2×C22⋊Q811yescharacteristic
229C23#57512yes—
230C23#57512yes—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—24no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—24no—
243C2×C4—24no—
244C2×C4—24no—
245C2×C4—24no—
246C2×C4—24no—
247C2×C4—24no—
248C2×C4—24no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—44no—
256C2×C4—44no—
257C2×C4—42no—
258C2×C4—44no—
259C2×C4—44no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—44no—
263C2×C4—44no—
264C2×C4—42no—
265C2×C4—44no—
266C2×C4—42no—
267C2×C4—44no—
268C2×C4—44no—
269C2×C4—44no—
270C2×C4—44no—
271C2×C4—44no—
272C2×C4—44no—
273C2×C4—82no—
274C2×C4—82no—
275C2×C4—82no—
276C2×C4—82no—
277C2×C4—82no—
278C2×C4—82no—
279C2×C4—82no—
280C2×C4—82no—
281C2×C4#39114yes—
282C2×C4#39114yes—
283C2×C4C2×C4×Q811yescharacteristic
284C2×C4C2×C4×Q811yescharacteristic
285C2×C4C2×#12411yescharacteristic
286C2×C4C2×C4⋊Q811yescharacteristic
287C8—84no—
288C22—21no—
289C22—21no—
290C22—21no—
291C22—21no—
292C22—22no—
293C22—22no—
294C22—22no—
295C22—22no—
296C22C2×#39111yescharacteristic
297C22#288311yescharacteristic
298C22#288411yescharacteristic
299C22#289612yes—
300C22#290112yes—
301C22#308011yescharacteristic
302C22#308011yescharacteristic
303C22#308012yes—
304C4—24no—
305C4—24no—
306C4—21no—
307C4—21no—
308C4—21no—
309C4—21no—
310C4—44no—
311C4—44no—
312C4—44no—
313C4—82no—
314C4—84no—
315C4—84no—
316C4—84no—
317C2—21no—
318C2—21no—
319C2—22no—
320C2#1762412yes—
321C2#1762912yes—
322C2#2131411yescharacteristic
323C2#2133011yescharacteristic
324C2#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
#219552
#138062
#99132
#99102
#80222
#80182