Finite Group Databasebeta

← #110201

Subgroups — #110201

Order 512.

746 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21844C211yesmaximal, characteristic
2#21407C211yesmaximal, characteristic
3#20331C211yesmaximal, characteristic
4#17501C211yesmaximal, characteristic
5#7974C211yesmaximal, characteristic
6#7940C211yesmaximal, characteristic
7#7861C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9#2920C2211yescharacteristic
10#2920C2211yescharacteristic
11C2×#405—22no—
12C2×C4×M16C2211yescharacteristic
13C22×#97C2211yescharacteristic
14#2283C2211yescharacteristic
15C4×#117C2211yescharacteristic
16#1994—22no—
17#1993C412yescomplemented
18#1986C414yescomplemented
19#1960C412yescomplemented
20#1892—22no—
21#1826—22no—
22#1763—22no—
23#1726—22no—
24C4×#97—22no—
25#1493—22no—
26#1491—22no—
27#1485C414yescomplemented
28#1288—22no—
29#1284—22no—
30#1275—22no—
31C22×M16—21no—
32C23×C8D412yes—
33#533—21no—
34#533—21no—
35#533—21no—
36#533—21no—
37#533—21no—
38#533—21no—
39#533C2×C412yes—
40#533C2×C412yes—
41C2×C4⋊Q8C2×C411yescharacteristic
42C2×C4⋊Q8C2×C411yescharacteristic
43C2×#125—21no—
44C2×C22⋊Q8—21no—
45C2×C22⋊Q8—21no—
46C2×C4×Q8—21no—
47C2×C4×Q8—21no—
48C2×#117—21no—
49C2×#117—21no—
50C22×C42C2311yescharacteristic
51#432—21no—
52#432—21no—
53#432—21no—
54#432—21no—
55#431—21no—
56#431—21no—
57#431—21no—
58#431C2×C412yes—
59#423—21no—
60#423—21no—
61C2×#103—24no—
62#407—21no—
63#407—21no—
64#405—216no—
65C4×M16—21no—
66C4×M16—21no—
67C4×M16—21no—
68C4×M16C2×C412yes—
69C4×M16C2×C412yes—
70C4×M16D412yes—
71C2×#97—24no—
72C2×#97—21no—
73C2×#97—21no—
74C2×#97—21no—
75C2×#97—21no—
76C2×#97—21no—
77C2×#97C2×C412yes—
78C2×#97D412yes—
79C2×C4×C8—24no—
80C2×C4×C8D412yes—
81#383—21no—
82#382C2×C412yes—
83#381—21no—
84#380—21no—
85C4×C4⋊C4—21no—
86C4×C4⋊C4—21no—
87C4×C4⋊C4C2×C412yes—
88C4×C22⋊C4—21no—
89C4×C22⋊C4—21no—
90C4×C22⋊C4D412yes—
91#376—21no—
92#376—21no—
93C43D412yes—
94#340—24no—
95#339—28no—
96Q8⋊C8—28no—
97C22×Q8—21no—
98C22×Q8—21no—
99C23×C4C2×D411yescharacteristic
100C23×C4C2×D411yescharacteristic
101C23×C4#4111yescharacteristic
102C2×M16—28no—
103C2×M16—22no—
104C2×M16—22no—
105C2×M16—22no—
106C2×M16—41no—
107C22×C8—22no—
108C22×C8—22no—
109C22×C8—22no—
110C22×C8—28no—
111C22×C8C22⋊C414yes—
112C4⋊Q8—21no—
113C4⋊Q8—21no—
114C4⋊Q8—21no—
115C4⋊Q8—21no—
116C4⋊Q8—21no—
117C4⋊Q8—21no—
118C4⋊Q8C4212yes—
119C4⋊Q8C4212yes—
120#125—22no—
121#125—22no—
122#125—41no—
123#125—41no—
124C22⋊Q8—22no—
125C22⋊Q8—22no—
126C22⋊Q8—22no—
127C22⋊Q8—22no—
128C22⋊Q8—41no—
129C22⋊Q8—41no—
130C22⋊Q8—41no—
131C22⋊Q8—41no—
132C4×Q8—22no—
133C4×Q8—22no—
134C4×Q8—22no—
135C4×Q8—22no—
136C4×Q8—41no—
137C4×Q8—41no—
138C4×Q8—41no—
139C4×Q8—41no—
140#117—28no—
141#117—22no—
142#117—22no—
143#117—41no—
144#117—41no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—41no—
152C2×C4⋊C4—41no—
153C2×C22⋊C4—22no—
154C2×C22⋊C4—41no—
155C2×C22⋊C4—41no—
156C2×C42—22no—
157C2×C42—22no—
158C2×C42—41no—
159C2×C42—41no—
160C2×C42C22⋊C414yes—
161C2×C42C22×C411yescharacteristic
162C2×C42C22×C411yescharacteristic
163C2×C42C22×C411yescharacteristic
164C2×C42C2×D411yescharacteristic
165C2×C42C2×D411yescharacteristic
166C2×C42C2×D411yescharacteristic
167C2×C42C2×D411yescharacteristic
168C2×C42#4111yescharacteristic
169C2×C42#4111yescharacteristic
170C2×C42#4111yescharacteristic
171C2×C42#4111yescharacteristic
172C2×C42#4111yescharacteristic
173C4⋊C8—22no—
174C4⋊C8—41no—
175C4⋊C8—41no—
176C4⋊C8—41no—
177#103—48no—
178C22⋊C8—22no—
179C22⋊C8—41no—
180C22⋊C8—41no—
181C22⋊C8—41no—
182#97—216no—
183#97—22no—
184#97—22no—
185#97—22no—
186#97—22no—
187#97—22no—
188#97—22no—
189#97—22no—
190#97—41no—
191C4×C8—216no—
192C4×C8—22no—
193C4×C8—22no—
194C4×C8—22no—
195C4×C8C22⋊C414yes—
196#95—22no—
197#95—22no—
198#95—22no—
199#95—22no—
200#95—41no—
201#95—41no—
202C24#12111yescharacteristic
203C2×Q8—22no—
204C2×Q8—22no—
205C2×Q8—41no—
206C2×Q8—41no—
207C2×Q8—44no—
208C2×Q8—82no—
209C22×C4—24no—
210C22×C4—24no—
211C22×C4—42no—
212C22×C4—41no—
213C22×C4—41no—
214C22×C4—42no—
215C22×C4—81no—
216C22×C4C2×C22⋊C411yescharacteristic
217C22×C4C2×C22⋊C411yescharacteristic
218C22×C4#11711yescharacteristic
219C22×C4C4×D411yescharacteristic
220C22×C4C4×D411yescharacteristic
221C22×C4C4×D411yescharacteristic
222C22×C4C4×D411yescharacteristic
223C22×C4C4×D411yescharacteristic
224C22×C4C4×D411yescharacteristic
225C22×C4#12011yescharacteristic
226C22×C4#12111yescharacteristic
227C22×C4#12111yescharacteristic
228C22×C4#12311yescharacteristic
229C22×C4#12311yescharacteristic
230C22×C4#12411yescharacteristic
231C22×C4#12411yescharacteristic
232C22×C4#12611yescharacteristic
233C22×C4#12711yescharacteristic
234M16—28no—
235M16—28no—
236M16—44no—
237M16—82no—
238C2×C8—24no—
239C2×C8—28no—
240C2×C8—24no—
241C2×C8—28no—
242C2×C8—24no—
243C2×C8—24no—
244C2×C8—24no—
245C2×C8—42no—
246C2×C8—42no—
247C2×C8#10418yes—
248C4⋊C4—28no—
249C4⋊C4—22no—
250C4⋊C4—22no—
251C4⋊C4—22no—
252C4⋊C4—22no—
253C4⋊C4—41no—
254C4⋊C4—42no—
255C4⋊C4—41no—
256C4⋊C4—41no—
257C4⋊C4—41no—
258C4⋊C4—42no—
259C4⋊C4—42no—
260C4⋊C4—42no—
261C4⋊C4—44no—
262C22⋊C4—28no—
263C22⋊C4—42no—
264C22⋊C4—42no—
265C22⋊C4—82no—
266C42—28no—
267C42—44no—
268C42—42no—
269C42—42no—
270C42—82no—
271C42—82no—
272C42C2×C4211yescharacteristic
273C42C2×C22⋊C411yescharacteristic
274C42#11711yescharacteristic
275C42#11711yescharacteristic
276C42C4×D411yescharacteristic
277C42C4×D411yescharacteristic
278C42C4×D411yescharacteristic
279C42C4×D411yescharacteristic
280C42C4×D411yescharacteristic
281C42C4×D411yescharacteristic
282C42#12111yescharacteristic
283C42#12111yescharacteristic
284C42#12111yescharacteristic
285C42#12311yescharacteristic
286C42#12411yescharacteristic
287C42#12611yescharacteristic
288C23—22no—
289C23—22no—
290C23—41no—
291C23#38611yescharacteristic
292C23#38811yescharacteristic
293C23#39011yescharacteristic
294Q8—44no—
295Q8—44no—
296Q8—82no—
297Q8—82no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—28no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—44no—
317C2×C4—44no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—82no—
321C2×C4—82no—
322C2×C4—82no—
323C2×C4—82no—
324C2×C4—82no—
325C2×C4C4×C22⋊C411yescharacteristic
326C2×C4C4×C22⋊C411yescharacteristic
327C2×C4C4×C22⋊C411yescharacteristic
328C2×C4#38611yescharacteristic
329C2×C4#38611yescharacteristic
330C2×C4#38811yescharacteristic
331C2×C4#38811yescharacteristic
332C2×C4#38811yescharacteristic
333C2×C4#38811yescharacteristic
334C2×C4#38811yescharacteristic
335C2×C4#39011yescharacteristic
336C2×C4#39011yescharacteristic
337C2×C4C2×#10414yes—
338C8—28no—
339C8—28no—
340C8—82no—
341C8—82no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—44no—
350C22#143711yescharacteristic
351C22#190912yes—
352C22#196212yes—
353C4—21no—
354C4—21no—
355C4—21no—
356C4—21no—
357C4—21no—
358C4—21no—
359C4—21no—
360C4—21no—
361C4—21no—
362C4—21no—
363C4—48no—
364C4—82no—
365C4—82no—
366C4—162no—
367C4C4×#10412yes—
368C4#189612yes—
369C2—22no—
370C2—22no—
371C2—41no—
372C2#1092311yescharacteristic
373C2#1094711yescharacteristic
374C2#1095011yescharacteristic
subgroupG/H
center, socleC22#1437
commutatorC42C2×C42
Frattini, cosocleC22×C42C23
prime radicalC24#121
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
17. #1993C4semidirect
18. #1986C4semidirect
19. #1960C4semidirect
27. #1485C4semidirect

Maximal subgroups

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

subgroupindex
#218442
#214072
#203312
#175012
#79742
#79402
#78612