Finite Group Databasebeta

← #117666

Subgroups — #117666

Order 512.

529 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21837C211yesmaximal, characteristic
2#17815C211yesmaximal, characteristic
3#17562C211yesmaximal, characteristic
4#17496C211yesmaximal, characteristic
5#7977C211yesmaximal, characteristic
6#7944C211yesmaximal, characteristic
7#7928C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9#2920C2211yescharacteristic
10#2920C2211yescharacteristic
11C2×#432C2211yescharacteristic
12C2×#431C2211yescharacteristic
13#2283C2211yescharacteristic
14#2283C2211yescharacteristic
15#1991—21no—
16#1991—21no—
17#1988C412yescomplemented
18#1987C412yescomplemented
19#1825—22no—
20#1572—22no—
21#1572—22no—
22#1567—21no—
23#1566—21no—
24#1515—22no—
25#1511—22no—
26#1503—22no—
27#1497—22no—
28#1491—22no—
29#1481—22no—
30#1288—22no—
31#1284—22no—
32#1282—22no—
33#533—21no—
34#533—21no—
35#533—21no—
36#533—21no—
37#533—21no—
38#533—21no—
39#533D412yes—
40#533Q812yes—
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—
49C22×C42C2311yescharacteristic
50#432—21no—
51#432—21no—
52#432—21no—
53#432—21no—
54#432—21no—
55#432—21no—
56#432—21no—
57#432D412yes—
58#431—21no—
59#431—21no—
60#431—21no—
61#431—21no—
62#431—21no—
63#431—21no—
64#431D412yes—
65#431D412yes—
66#423—21no—
67#423—21no—
68C2×C4⋊C8—21no—
69C2×C4⋊C8C2×C412yes—
70#407—21no—
71#407—21no—
72C2×C22⋊C8—21no—
73C2×C22⋊C8D412yes—
74C4×M16—21no—
75C4×M16—21no—
76C2×#97—21no—
77C2×C4×C8D411yescharacteristic
78C2×C4×C8D411yescharacteristic
79#383—21no—
80#383—21no—
81#382—21no—
82#382—21no—
83#381—21no—
84#381D412yes—
85#380—21no—
86#380—21no—
87C4×C4⋊C4—21no—
88C4×C4⋊C4C2×C412yes—
89C4×C22⋊C4—21no—
90C4×C22⋊C4—21no—
91#376—21no—
92#376—21no—
93#340—22no—
94#340—22no—
95#340—22no—
96#340—22no—
97#335—24no—
98#334—24no—
99#328—28no—
100Q8⋊C8—28no—
101C22×Q8—21no—
102C22×Q8—21no—
103C23×C4C2×D411yescharacteristic
104C23×C4C2×D411yescharacteristic
105C23×C4#4111yescharacteristic
106C2×M16—41no—
107C2×M16—41no—
108C22×C8—41no—
109C22×C8C22⋊C412yes—
110C22×C8C22⋊C412yes—
111C4⋊Q8—21no—
112C4⋊Q8—21no—
113C4⋊Q8—21no—
114C4⋊Q8—21no—
115C4⋊Q8—21no—
116C4⋊Q8—21no—
117C4⋊Q8C4⋊C412yes—
118C4⋊Q8C4⋊C412yes—
119#125—22no—
120#125—22no—
121#125—41no—
122#125—41no—
123C22⋊Q8—22no—
124C22⋊Q8—22no—
125C22⋊Q8—22no—
126C22⋊Q8—22no—
127C22⋊Q8—41no—
128C22⋊Q8—41no—
129C22⋊Q8—41no—
130C22⋊Q8—41no—
131C4×Q8—22no—
132C4×Q8—22no—
133C4×Q8—22no—
134C4×Q8—22no—
135C4×Q8—41no—
136C4×Q8—41no—
137C4×Q8—41no—
138C4×Q8—41no—
139#117—22no—
140#117—22no—
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—41no—
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—41no—
156C2×C42—41no—
157C2×C42C22×C411yescharacteristic
158C2×C42C2×D411yescharacteristic
159C2×C42C2×D411yescharacteristic
160C2×C42C2×D411yescharacteristic
161C2×C42C2×D411yescharacteristic
162C2×C42C2×D411yescharacteristic
163C2×C42C2×Q811yescharacteristic
164C2×C42#4111yescharacteristic
165C2×C42#4111yescharacteristic
166C2×C42#4111yescharacteristic
167C2×C42#4111yescharacteristic
168C2×C42#4111yescharacteristic
169C4⋊C8—22no—
170C4⋊C8—22no—
171C4⋊C8—22no—
172C4⋊C8—22no—
173C4⋊C8—22no—
174C4⋊C8—41no—
175C4⋊C8—41no—
176C4⋊C8—41no—
177C22⋊C8—22no—
178C22⋊C8—22no—
179C22⋊C8—22no—
180C22⋊C8—22no—
181C22⋊C8—22no—
182C22⋊C8—41no—
183C22⋊C8—41no—
184C22⋊C8—41no—
185#97—22no—
186#97—41no—
187#97—41no—
188#97—41no—
189C4×C8—21no—
190C4×C8—21no—
191C4×C8—22no—
192C4×C8—22no—
193C4×C8—21no—
194C4×C8—21no—
195#95—22no—
196#95—22no—
197#95—22no—
198#95—41no—
199#95—41no—
200#95—41no—
201#95—41no—
202#95—41no—
203C24#12111yescharacteristic
204C2×Q8—22no—
205C2×Q8—22no—
206C2×Q8—41no—
207C2×Q8—41no—
208C2×Q8—82no—
209C2×Q8—82no—
210C22×C4—42no—
211C22×C4—42no—
212C22×C4—41no—
213C22×C4—41no—
214C22×C4—42no—
215C22×C4—42no—
216C22×C4—81no—
217C22×C4C2×C22⋊C411yescharacteristic
218C22×C4C4×D411yescharacteristic
219C22×C4C4×D411yescharacteristic
220C22×C4#12011yescharacteristic
221C22×C4#12111yescharacteristic
222C22×C4#12111yescharacteristic
223C22×C4#12111yescharacteristic
224C22×C4#12111yescharacteristic
225C22×C4#12111yescharacteristic
226C22×C4C22⋊Q811yescharacteristic
227C22×C4C22⋊Q811yescharacteristic
228C22×C4C22⋊Q811yescharacteristic
229C22×C4#12311yescharacteristic
230C22×C4#12311yescharacteristic
231C22×C4#12411yescharacteristic
232C22×C4#12411yescharacteristic
233C22×C4#12611yescharacteristic
234C22×C4#12711yescharacteristic
235M16—82no—
236M16—82no—
237C2×C8—22no—
238C2×C8—22no—
239C2×C8—42no—
240C2×C8—42no—
241C2×C8—42no—
242C2×C8—42no—
243C2×C8—42no—
244C2×C8—82no—
245C4⋊C4—22no—
246C4⋊C4—22no—
247C4⋊C4—22no—
248C4⋊C4—22no—
249C4⋊C4—44no—
250C4⋊C4—41no—
251C4⋊C4—41no—
252C4⋊C4—41no—
253C4⋊C4—41no—
254C4⋊C4—42no—
255C4⋊C4—42no—
256C4⋊C4—42no—
257C4⋊C4—42no—
258C4⋊C4—82no—
259C22⋊C4—42no—
260C22⋊C4—42no—
261C22⋊C4—44no—
262C22⋊C4—82no—
263C42—42no—
264C42—42no—
265C42—82no—
266C42—82no—
267C42C2×C4⋊C411yescharacteristic
268C42#11711yescharacteristic
269C42C4×D411yescharacteristic
270C42C4×D411yescharacteristic
271C42#12011yescharacteristic
272C42#12011yescharacteristic
273C42#12111yescharacteristic
274C42#12111yescharacteristic
275C42C22⋊Q811yescharacteristic
276C42C22⋊Q811yescharacteristic
277C42C22⋊Q811yescharacteristic
278C42#12311yescharacteristic
279C42#12311yescharacteristic
280C42#12311yescharacteristic
281C42#12411yescharacteristic
282C42#12611yescharacteristic
283C23—22no—
284C23—41no—
285C23—41no—
286C23#39011yescharacteristic
287C23#39411yescharacteristic
288C23#39611yescharacteristic
289Q8—44no—
290Q8—82no—
291Q8—82no—
292Q8—82no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—41no—
299C2×C4—41no—
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—41no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—82no—
314C2×C4—82no—
315C2×C4—82no—
316C2×C4—82no—
317C2×C4—82no—
318C2×C4—82no—
319C2×C4—82no—
320C2×C4—82no—
321C2×C4#38011yescharacteristic
322C2×C4#38511yescharacteristic
323C2×C4#38511yescharacteristic
324C2×C4#38611yescharacteristic
325C2×C4#38811yescharacteristic
326C2×C4#38811yescharacteristic
327C2×C4#39211yescharacteristic
328C2×C4#39311yescharacteristic
329C2×C4#39411yescharacteristic
330C2×C4#39611yescharacteristic
331C2×C4#39711yescharacteristic
332C2×C4#39911yescharacteristic
333C8—42no—
334C8—42no—
335C8—82no—
336C8—82no—
337C8—82no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—82no—
347C22#143911yescharacteristic
348C22#197612yes—
349C4—21no—
350C4—21no—
351C4—21no—
352C4—21no—
353C4—21no—
354C4—21no—
355C4—21no—
356C4—21no—
357C4—21no—
358C4—21no—
359C4—21no—
360C4—82no—
361C4—82no—
362C4—162no—
363C4—162no—
364C4#190912yes—
365C2—22no—
366C2—41no—
367C2—41no—
368C2#1107411yescharacteristic
369C2#1112611yescharacteristic
370C2#1113111yescharacteristic
subgroupG/H
center, socleC22#1439
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#121
LCSG ▷ C2×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. #1988C4semidirect
18. #1987C4semidirect

Maximal subgroups

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

subgroupindex
#218372
#178152
#175622
#174962
#79772
#79442
#79282