Finite Group Databasebeta

← #117488

Subgroups — #117488

Order 512.

522 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17816C211yesmaximal, characteristic
2#17715C211yesmaximal, characteristic
3#17596C211yesmaximal, characteristic
4#17464C211yesmaximal, characteristic
5#8110C211yesmaximal, characteristic
6#8090C211yesmaximal, characteristic
7#7894C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9#2920C2211yescharacteristic
10#2920C2211yescharacteristic
11C2×#432C2211yescharacteristic
12C2×#431C2211yescharacteristic
13C2×#431C2211yescharacteristic
14#2283C2211yescharacteristic
15#1572—22no—
16#1571—21no—
17#1570—21no—
18#1568—22no—
19#1552—22no—
20#1549C412yescomplemented
21#1546C412yescomplemented
22#1544—21no—
23#1544—21no—
24#1520—22no—
25#1515—22no—
26#1506—22no—
27#1485—22no—
28#1481—22no—
29#1479—22no—
30#1305—22no—
31#1302—22no—
32#1279—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#431—21no—
65#431—21no—
66#431C2×C412yes—
67#431D412yes—
68#431D412yes—
69#431D411yescharacteristic
70#431D411yescharacteristic
71#423—21no—
72#423C2×C412yes—
73C2×C4⋊C8—21no—
74C2×C4⋊C8—21no—
75C2×C4⋊C8D412yes—
76#407—21no—
77#407—21no—
78C2×C22⋊C8—21no—
79C2×C22⋊C8—21no—
80C2×C22⋊C8—21no—
81C4×M16—21no—
82C4×M16—21no—
83C2×#97—21no—
84C2×C4×C8—21no—
85C2×C4×C8—21no—
86#383—21no—
87#382—21no—
88#381—21no—
89#380D412yes—
90C4×C4⋊C4—21no—
91C4×C22⋊C4—21no—
92#376—21no—
93#335—24no—
94#334—24no—
95#332—22no—
96#332—22no—
97#332—22no—
98#332—22no—
99#328—28no—
100Q8⋊C8—28no—
101C22×Q8—21no—
102C22×Q8—21no—
103C23×C4C2×D411yescharacteristic
104C23×C4#4111yescharacteristic
105C23×C4#4111yescharacteristic
106C2×M16—22no—
107C2×M16—41no—
108C22×C8—22no—
109C22×C8—41no—
110C22×C8—41no—
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×C22⋊C4—22no—
151C2×C22⋊C4—41no—
152C2×C42—41no—
153C2×C42—41no—
154C2×C42C22×C411yescharacteristic
155C2×C42C2×D411yescharacteristic
156C2×C42C2×D411yescharacteristic
157C2×C42C2×D411yescharacteristic
158C2×C42C2×D411yescharacteristic
159C2×C42C2×D411yescharacteristic
160C2×C42C2×D411yescharacteristic
161C2×C42C2×Q811yescharacteristic
162C2×C42#4111yescharacteristic
163C2×C42#4111yescharacteristic
164C2×C42#4111yescharacteristic
165C2×C42#4111yescharacteristic
166C4⋊C8—22no—
167C4⋊C8—22no—
168C4⋊C8—22no—
169C4⋊C8—22no—
170C4⋊C8—22no—
171C4⋊C8—22no—
172C4⋊C8—41no—
173C4⋊C8—41no—
174C4⋊C8—41no—
175C4⋊C8C22⋊C412yes—
176C4⋊C8C22⋊C412yes—
177C22⋊C8—22no—
178C22⋊C8—21no—
179C22⋊C8—21no—
180C22⋊C8—22no—
181C22⋊C8—22no—
182C22⋊C8—22no—
183C22⋊C8—41no—
184C22⋊C8—41no—
185C22⋊C8—41no—
186C22⋊C8—41no—
187C22⋊C8—41no—
188#97—22no—
189#97—41no—
190#97—41no—
191#97—41no—
192C4×C8—22no—
193C4×C8—21no—
194C4×C8—21no—
195C4×C8—22no—
196C4×C8—22no—
197C4×C8—41no—
198C4×C8—41no—
199#95—22no—
200#95—41no—
201#95—41no—
202#95—41no—
203C24#12311yescharacteristic
204C2×Q8—22no—
205C2×Q8—22no—
206C2×Q8—44no—
207C2×Q8—41no—
208C2×Q8—41no—
209C2×Q8—82no—
210C22×C4—42no—
211C22×C4—42no—
212C22×C4—42no—
213C22×C4—41no—
214C22×C4—41no—
215C22×C4—42no—
216C22×C4#11711yescharacteristic
217C22×C4C4×D411yescharacteristic
218C22×C4C4×D411yescharacteristic
219C22×C4#12011yescharacteristic
220C22×C4#12111yescharacteristic
221C22×C4#12111yescharacteristic
222C22×C4#12111yescharacteristic
223C22×C4#12111yescharacteristic
224C22×C4C22⋊Q811yescharacteristic
225C22×C4C22⋊Q811yescharacteristic
226C22×C4C22⋊Q811yescharacteristic
227C22×C4#12311yescharacteristic
228C22×C4#12311yescharacteristic
229C22×C4#12311yescharacteristic
230C22×C4#12411yescharacteristic
231C22×C4#12411yescharacteristic
232C22×C4#12611yescharacteristic
233C22×C4#12711yescharacteristic
234M16—82no—
235M16—82no—
236C2×C8—22no—
237C2×C8—22no—
238C2×C8—42no—
239C2×C8—42no—
240C2×C8—42no—
241C2×C8—42no—
242C2×C8—82no—
243C2×C8—82no—
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—
258C22⋊C4—42no—
259C22⋊C4—42no—
260C22⋊C4—82no—
261C42—42no—
262C42—42no—
263C42—82no—
264C42C2×C22⋊C411yescharacteristic
265C42C2×C4⋊C411yescharacteristic
266C42C4×D411yescharacteristic
267C42C4×D411yescharacteristic
268C42#12011yescharacteristic
269C42#12011yescharacteristic
270C42#12111yescharacteristic
271C42#12111yescharacteristic
272C42#12111yescharacteristic
273C42#12111yescharacteristic
274C42C22⋊Q811yescharacteristic
275C42C22⋊Q811yescharacteristic
276C42C22⋊Q811yescharacteristic
277C42#12311yescharacteristic
278C42#12411yescharacteristic
279C42#12611yescharacteristic
280C23—22no—
281C23—41no—
282C23—41no—
283C23#38811yescharacteristic
284C23#39411yescharacteristic
285C23#39911yescharacteristic
286Q8—44no—
287Q8—82no—
288Q8—82no—
289Q8—82no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—22no—
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—42no—
309C2×C4—42no—
310C2×C4—82no—
311C2×C4—82no—
312C2×C4—82no—
313C2×C4—82no—
314C2×C4—82no—
315C2×C4#38011yescharacteristic
316C2×C4#38511yescharacteristic
317C2×C4#38511yescharacteristic
318C2×C4#38611yescharacteristic
319C2×C4#38811yescharacteristic
320C2×C4#39011yescharacteristic
321C2×C4#39211yescharacteristic
322C2×C4#39311yescharacteristic
323C2×C4#39411yescharacteristic
324C2×C4#39611yescharacteristic
325C2×C4#39611yescharacteristic
326C2×C4#39711yescharacteristic
327C8—42no—
328C8—42no—
329C8—82no—
330C8—82no—
331C8—82no—
332C8—82no—
333C22—21no—
334C22—21no—
335C22—21no—
336C22—21no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—82no—
342C22#143911yescharacteristic
343C22#204912yes—
344C4—21no—
345C4—21no—
346C4—21no—
347C4—21no—
348C4—21no—
349C4—21no—
350C4—21no—
351C4—21no—
352C4—21no—
353C4—21no—
354C4—82no—
355C4—82no—
356C4—162no—
357C4#188112yes—
358C4#197612yes—
359C2—22no—
360C2—41no—
361C2—41no—
362C2#1106511yescharacteristic
363C2#1112611yescharacteristic
364C2#1114311yescharacteristic
subgroupG/H
center, socleC22#1439
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#123
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
20. #1549C4semidirect
21. #1546C4semidirect

Maximal subgroups

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

subgroupindex
#178162
#177152
#175962
#174642
#81102
#80902
#78942