Finite Group Databasebeta

← #206885

Subgroups — #206885

Order 512.

966 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2330C211yesmaximal, characteristic
2#21951C211yesmaximal, characteristic
3#20536C211yesmaximal, characteristic
4#9883C214yesmaximal
5#9881C214yesmaximal
6#7908C212yesmaximal
7#7906C212yesmaximal
8C22×#125C411yescharacteristic
9C2×#431C2212yes—
10C2×#431C2212yes—
11#2330—24no—
12#2330—24no—
13C2×#389C412yes—
14C2×#389C411yescharacteristic
15C2×#389C411yescharacteristic
16C2×#389C2212yes—
17C2×#387C412yes—
18C2×#387C2212yes—
19C2×#384C411yescharacteristic
20C2×C4×C4⋊C4C2211yescharacteristic
21C2×C4×C4⋊C4C2211yescharacteristic
22C2×#376C2211yescharacteristic
23#1939—24no—
24#1797C2214yes—
25#1770—24no—
26#1749C2214yes—
27#1280C2214yes—
28#1280C2218yes—
29#1280C2214yes—
30C2×#125—24no—
31C2×#125—24no—
32C2×#125—24no—
33C22×C4⋊C4C2×C412yes—
34C22×C4⋊C4C2×C412yes—
35C22×C4⋊C4C2×C411yescharacteristic
36C22×C4⋊C4C2×C411yescharacteristic
37C22×C4⋊C4C2×C411yescharacteristic
38C22×C4⋊C4C2×C411yescharacteristic
39C22×C4⋊C4C2×C412yes—
40C22×C4⋊C4C2×C412yes—
41C22×C4⋊C4C2×C412yes—
42C22×C4⋊C4C2×C412yes—
43C22×C42C2×C411yescharacteristic
44C22×C42C2×C411yescharacteristic
45C22×C42C2×C412yes—
46C22×C42C2312yes—
47C22×C42C2311yescharacteristic
48#431—24no—
49#431—28no—
50#431Q818yes—
51C2×C4⋊C8—22no—
52C2×C4⋊C8—22no—
53C2×C4⋊C8—24no—
54C2×C22⋊C8C2314yes—
55C2×C22⋊C8C2314yes—
56C2×#97—24no—
57C2×C4×C8—22no—
58C2×C4×C8—22no—
59#389—24no—
60#389—24no—
61#389—24no—
62#389—24no—
63#389—28no—
64#387—24no—
65#387—24no—
66#387—28no—
67#384—24no—
68C4×C4⋊C4—22no—
69C4×C4⋊C4—22no—
70C4×C4⋊C4D414yes—
71C4×C4⋊C4D414yes—
72#376—22no—
73#376—22no—
74C2×#95C2×C411yescharacteristic
75C2×#95C2×C411yescharacteristic
76C2×#95C2×C412yes—
77C2×#95C2×C412yes—
78C2×#95C2×C412yes—
79C2×#95C2311yescharacteristic
80C2×#95C2311yescharacteristic
81C2×#95C2312yes—
82#336—24no—
83#336—24no—
84C23×C4C22×C411yescharacteristic
85C23×C4C22×C412yes—
86C23×C4C22×C411yescharacteristic
87C23×C4C22×C412yes—
88C23×C4C22×C412yes—
89C23×C4C22×C411yescharacteristic
90C23×C4C22×C411yescharacteristic
91C23×C4C22×C412yes—
92C23×C4C22×C411yescharacteristic
93C23×C4C22×C411yescharacteristic
94C23×C4C2411yescharacteristic
95C22×C8—24no—
96C22×C8—24no—
97#125—232no—
98C2×C4⋊C4—22no—
99C2×C4⋊C4—24no—
100C2×C4⋊C4—24no—
101C2×C4⋊C4—24no—
102C2×C4⋊C4—24no—
103C2×C4⋊C4—24no—
104C2×C4⋊C4—24no—
105C2×C4⋊C4—22no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4—24no—
109C2×C4⋊C4—22no—
110C2×C4⋊C4—24no—
111C2×C4⋊C4—24no—
112C2×C4⋊C4—24no—
113C2×C4⋊C4—24no—
114C2×C4⋊C4—24no—
115C2×C4⋊C4—82no—
116C2×C4⋊C4—82no—
117C2×C4⋊C4—82no—
118C2×C4⋊C4—82no—
119C2×C4⋊C4C22⋊C414yes—
120C2×C4⋊C4C22⋊C414yes—
121C2×C4⋊C4M1614yes—
122C2×C42—22no—
123C2×C42—22no—
124C2×C42—22no—
125C2×C42—24no—
126C2×C42—24no—
127C2×C42—82no—
128C2×C42—82no—
129C2×C42M1614yes—
130C2×C42C2×D414yes—
131C2×C42C2×D414yes—
132C2×C42C2×D412yes—
133C2×C42C2×D412yes—
134C2×C42C2×Q814yes—
135C2×C42#4114yes—
136C4⋊C8—28no—
137C4⋊C8—44no—
138C4⋊C8—84no—
139C22⋊C8—24no—
140C22⋊C8—28no—
141C22⋊C8C2×Q818yes—
142#97—84no—
143C4×C8—28no—
144C4×C8—44no—
145#95—22no—
146#95—22no—
147#95—24no—
148#95—24no—
149#95—21no—
150#95—21no—
151#95—21no—
152#95—21no—
153#95—22no—
154#95—22no—
155#95C22⋊C418yes—
156C24C23×C411yescharacteristic
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—22no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4—22no—
165C22×C4—24no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—22no—
171C22×C4—24no—
172C22×C4—41no—
173C22×C4—41no—
174C22×C4—42no—
175C22×C4—41no—
176C22×C4—42no—
177C22×C4—41no—
178C22×C4—42no—
179C22×C4—42no—
180C22×C4—42no—
181C22×C4—41no—
182C22×C4—41no—
183C22×C4—41no—
184C22×C4—42no—
185C22×C4—41no—
186C22×C4—42no—
187C22×C4—41no—
188C22×C4—42no—
189C22×C4—41no—
190C22×C4—41no—
191C22×C4—41no—
192C22×C4C2×C22⋊C414yes—
193C22×C4C2×C22⋊C414yes—
194C22×C4C2×C22⋊C412yes—
195C22×C4C2×C22⋊C412yes—
196C22×C4C4×Q814yes—
197C22×C4C4×Q814yes—
198C22×C4C2×M1612yes—
199C22×C4C2×M1612yes—
200C22×C4C2×M1614yes—
201C22×C4C2×M1612yes—
202C22×C4C2×M1612yes—
203C22×C4#13114yes—
204C22×C4C22×D411yescharacteristic
205C22×C4C22×D411yescharacteristic
206C22×C4C22×Q812yes—
207C22×C4C2×#4112yes—
208C2×C8—28no—
209C2×C8—44no—
210C2×C8—44no—
211C2×C8—44no—
212C2×C8—44no—
213C2×C8—44no—
214C4⋊C4—216no—
215C4⋊C4—216no—
216C4⋊C4—28no—
217C4⋊C4—28no—
218C4⋊C4—28no—
219C4⋊C4—44no—
220C4⋊C4—84no—
221C4⋊C4—84no—
222C4⋊C4—84no—
223C4⋊C4—84no—
224C42—28no—
225C42—44no—
226C42—44no—
227C42—44no—
228C42—84no—
229C42C22⋊Q814yes—
230C42C22⋊Q814yes—
231C42#12414yes—
232C42C4⋊Q814yes—
233C23—21no—
234C23—21no—
235C23—21no—
236C23—21no—
237C23C22×C22⋊C411yescharacteristic
238C23C2×C4×Q812yes—
239C23C22×M1611yescharacteristic
240C23C22×M1611yescharacteristic
241C23C2×#13112yes—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—24no—
251C2×C4—24no—
252C2×C4—24no—
253C2×C4—24no—
254C2×C4—24no—
255C2×C4—24no—
256C2×C4—24no—
257C2×C4—44no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—44no—
261C2×C4—44no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—44no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—42no—
272C2×C4—44no—
273C2×C4—42no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—42no—
277C2×C4—41no—
278C2×C4—42no—
279C2×C4—41no—
280C2×C4—42no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—84no—
284C2×C4—82no—
285C2×C4—84no—
286C2×C4—82no—
287C2×C4#39114yes—
288C2×C4#39114yes—
289C2×C4#40714yes—
290C2×C4#40814yes—
291C2×C4#44618yes—
292C2×C4#44618yes—
293C2×C4C2×C22⋊Q812yes—
294C2×C4C2×#12411yescharacteristic
295C2×C4C2×C4⋊Q811yescharacteristic
296C8—48no—
297C8—84no—
298C22—22no—
299C22—22no—
300C22—21no—
301C22—21no—
302C22—21no—
303C22—21no—
304C22—22no—
305C22—22no—
306C22C2×#39111yescharacteristic
307C22C2×#40711yescharacteristic
308C22C2×#40811yescharacteristic
309C22#288311yescharacteristic
310C22#288411yescharacteristic
311C22#288811yescharacteristic
312C22#288911yescharacteristic
313C22C2×#44612yes—
314C22C2×#44612yes—
315C4—22no—
316C4—21no—
317C4—21no—
318C4—24no—
319C4—44no—
320C4—42no—
321C4—42no—
322C4—44no—
323C4—44no—
324C4—81no—
325C4—82no—
326C4—81no—
327C4#198518yes—
328C2—22no—
329C2—22no—
330C2#1737011yescharacteristic
331C2#1737211yescharacteristic
332C2#1738911yescharacteristic
333C2#1739111yescharacteristic
334C2C2×#198511yescharacteristic
335C2#2133411yescharacteristic
336C2#2133811yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutator, prime radicalC24C23×C4
Frattini, cosocleC23×C4C24
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Maximal subgroups

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

subgroupindex
C2×#23302
#219512
#205362
#98832
#98812
#79082
#79062