Finite Group Databasebeta

← #110203

Subgroups — #110203

Order 512.

634 conjugacy classes of subgroups.

The subgroup lattice has 371 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21844C211yesmaximal, characteristic
2#21407C211yesmaximal, characteristic
3#20367C211yesmaximal, characteristic
4#17501C211yesmaximal, characteristic
5#7974C211yesmaximal, characteristic
6#7940C211yesmaximal, characteristic
7#7861C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9#2920C2211yescharacteristic
10#2920C2211yescharacteristic
11C2×C4×M16C2211yescharacteristic
12C22×#97C2211yescharacteristic
13#2283C2211yescharacteristic
14C4×#117C2211yescharacteristic
15#1994—22no—
16#1993C412yescomplemented
17#1986C414yescomplemented
18#1960C412yescomplemented
19#1892—22no—
20#1826—22no—
21#1763—22no—
22#1746—22no—
23#1745—22no—
24#1729—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—
64C4×M16—21no—
65C4×M16—21no—
66C4×M16—21no—
67C4×M16C2×C412yes—
68C4×M16C2×C412yes—
69C4×M16D412yes—
70C2×#97—21no—
71C2×#97—21no—
72C2×#97—21no—
73C2×#97—21no—
74C2×#97—21no—
75C2×#97C2×C412yes—
76C2×#97D412yes—
77C2×C4×C8D412yes—
78#383—21no—
79#382C2×C412yes—
80#381—21no—
81#380—21no—
82C4×C4⋊C4—21no—
83C4×C4⋊C4—21no—
84C4×C4⋊C4C2×C412yes—
85C4×C22⋊C4—21no—
86C4×C22⋊C4—21no—
87C4×C22⋊C4D412yes—
88#376—21no—
89#376—21no—
90C43D412yes—
91#340—24no—
92#339—28no—
93#336—28no—
94Q8⋊C8—28no—
95C22×Q8—21no—
96C22×Q8—21no—
97C23×C4C2×D411yescharacteristic
98C23×C4C2×D411yescharacteristic
99C23×C4#4111yescharacteristic
100C2×M16—22no—
101C2×M16—22no—
102C2×M16—22no—
103C2×M16—44no—
104C2×M16—41no—
105C22×C8—22no—
106C22×C8—22no—
107C22×C8—22no—
108C22×C8—44no—
109C22×C8C22⋊C414yes—
110C4⋊Q8—21no—
111C4⋊Q8—21no—
112C4⋊Q8—21no—
113C4⋊Q8—21no—
114C4⋊Q8—21no—
115C4⋊Q8—21no—
116C4⋊Q8C4212yes—
117C4⋊Q8C4212yes—
118#125—22no—
119#125—22no—
120#125—41no—
121#125—41no—
122C22⋊Q8—22no—
123C22⋊Q8—22no—
124C22⋊Q8—22no—
125C22⋊Q8—22no—
126C22⋊Q8—41no—
127C22⋊Q8—41no—
128C22⋊Q8—41no—
129C22⋊Q8—41no—
130C4×Q8—22no—
131C4×Q8—22no—
132C4×Q8—22no—
133C4×Q8—22no—
134C4×Q8—41no—
135C4×Q8—41no—
136C4×Q8—41no—
137C4×Q8—41no—
138#117—22no—
139#117—22no—
140#117—44no—
141#117—41no—
142#117—41no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—41no—
150C2×C4⋊C4—41no—
151C2×C22⋊C4—22no—
152C2×C22⋊C4—41no—
153C2×C22⋊C4—41no—
154C2×C42—22no—
155C2×C42—22no—
156C2×C42—41no—
157C2×C42—41no—
158C2×C42C22⋊C414yes—
159C2×C42C22×C411yescharacteristic
160C2×C42C22×C411yescharacteristic
161C2×C42C22×C411yescharacteristic
162C2×C42C2×D411yescharacteristic
163C2×C42C2×D411yescharacteristic
164C2×C42C2×D411yescharacteristic
165C2×C42C2×D411yescharacteristic
166C2×C42#4111yescharacteristic
167C2×C42#4111yescharacteristic
168C2×C42#4111yescharacteristic
169C2×C42#4111yescharacteristic
170C2×C42#4111yescharacteristic
171C4⋊C8—22no—
172C4⋊C8—41no—
173C4⋊C8—41no—
174C4⋊C8—41no—
175#103—48no—
176C22⋊C8—22no—
177C22⋊C8—41no—
178C22⋊C8—41no—
179C22⋊C8—41no—
180#97—22no—
181#97—22no—
182#97—22no—
183#97—22no—
184#97—22no—
185#97—22no—
186#97—22no—
187#97—41no—
188C4×C8—22no—
189C4×C8—22no—
190C4×C8—22no—
191C4×C8C22⋊C414yes—
192#95—22no—
193#95—22no—
194#95—22no—
195#95—22no—
196#95—41no—
197#95—41no—
198C24#12111yescharacteristic
199C2×Q8—22no—
200C2×Q8—22no—
201C2×Q8—41no—
202C2×Q8—41no—
203C2×Q8—44no—
204C2×Q8—82no—
205C22×C4—24no—
206C22×C4—24no—
207C22×C4—42no—
208C22×C4—41no—
209C22×C4—41no—
210C22×C4—42no—
211C22×C4—81no—
212C22×C4C2×C22⋊C411yescharacteristic
213C22×C4C2×C22⋊C411yescharacteristic
214C22×C4#11711yescharacteristic
215C22×C4C4×D411yescharacteristic
216C22×C4C4×D411yescharacteristic
217C22×C4C4×D411yescharacteristic
218C22×C4C4×D411yescharacteristic
219C22×C4C4×D411yescharacteristic
220C22×C4C4×D411yescharacteristic
221C22×C4#12011yescharacteristic
222C22×C4#12111yescharacteristic
223C22×C4#12111yescharacteristic
224C22×C4#12311yescharacteristic
225C22×C4#12311yescharacteristic
226C22×C4#12411yescharacteristic
227C22×C4#12411yescharacteristic
228C22×C4#12611yescharacteristic
229C22×C4#12711yescharacteristic
230M16—44no—
231M16—44no—
232M16—44no—
233M16—82no—
234C2×C8—24no—
235C2×C8—24no—
236C2×C8—24no—
237C2×C8—24no—
238C2×C8—44no—
239C2×C8—44no—
240C2×C8—44no—
241C2×C8—42no—
242C2×C8—42no—
243C4⋊C4—22no—
244C4⋊C4—22no—
245C4⋊C4—22no—
246C4⋊C4—22no—
247C4⋊C4—44no—
248C4⋊C4—41no—
249C4⋊C4—42no—
250C4⋊C4—41no—
251C4⋊C4—41no—
252C4⋊C4—41no—
253C4⋊C4—42no—
254C4⋊C4—42no—
255C4⋊C4—42no—
256C4⋊C4—44no—
257C22⋊C4—44no—
258C22⋊C4—42no—
259C22⋊C4—42no—
260C22⋊C4—82no—
261C42—44no—
262C42—44no—
263C42—42no—
264C42—42no—
265C42—82no—
266C42—82no—
267C42C2×C4211yescharacteristic
268C42C2×C22⋊C411yescharacteristic
269C42#11711yescharacteristic
270C42#11711yescharacteristic
271C42C4×D411yescharacteristic
272C42C4×D411yescharacteristic
273C42C4×D411yescharacteristic
274C42C4×D411yescharacteristic
275C42C4×D411yescharacteristic
276C42C4×D411yescharacteristic
277C42#12111yescharacteristic
278C42#12111yescharacteristic
279C42#12111yescharacteristic
280C42#12311yescharacteristic
281C42#12411yescharacteristic
282C42#12611yescharacteristic
283C23—22no—
284C23—22no—
285C23—41no—
286C23#38611yescharacteristic
287C23#38811yescharacteristic
288C23#39011yescharacteristic
289Q8—44no—
290Q8—44no—
291Q8—82no—
292Q8—82no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—44no—
311C2×C4—44no—
312C2×C4—44no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—82no—
316C2×C4—82no—
317C2×C4—82no—
318C2×C4—82no—
319C2×C4—82no—
320C2×C4C4×C22⋊C411yescharacteristic
321C2×C4C4×C22⋊C411yescharacteristic
322C2×C4C4×C22⋊C411yescharacteristic
323C2×C4#38611yescharacteristic
324C2×C4#38611yescharacteristic
325C2×C4#38811yescharacteristic
326C2×C4#38811yescharacteristic
327C2×C4#38811yescharacteristic
328C2×C4#38811yescharacteristic
329C2×C4#38811yescharacteristic
330C2×C4#39011yescharacteristic
331C2×C4#39011yescharacteristic
332C2×C4#42114yes—
333C8—44no—
334C8—44no—
335C8—82no—
336C8—82no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—44no—
345C22#143711yescharacteristic
346C22#190912yes—
347C22#196312yes—
348C4—21no—
349C4—21no—
350C4—21no—
351C4—21no—
352C4—21no—
353C4—21no—
354C4—21no—
355C4—21no—
356C4—21no—
357C4—21no—
358C4—84no—
359C4—82no—
360C4—82no—
361C4—162no—
362C4#175912yes—
363C4#189712yes—
364C2—22no—
365C2—22no—
366C2—41no—
367C2#1092411yescharacteristic
368C2#1094711yescharacteristic
369C2#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
16. #1993C4semidirect
17. #1986C4semidirect
18. #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
#203672
#175012
#79742
#79402
#78612