Finite Group Databasebeta

← #110692

Subgroups — #110692

Order 512.

570 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21845C211yesmaximal, characteristic
2#17734C211yesmaximal, characteristic
3C2×#1485C211yesmaximal, characteristic
4#17334C211yesmaximal, characteristic
5#8058C211yesmaximal, characteristic
6#7972C211yesmaximal, characteristic
7#7956C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9#2920C2211yescharacteristic
10#2920C2211yescharacteristic
11C2×#432C2211yescharacteristic
12C2×#431C2211yescharacteristic
13C2×C4×M16C2211yescharacteristic
14#2283C2211yescharacteristic
15#1995C414yes—
16#1986—22no—
17#1826—22no—
18#1559—22no—
19#1558C412yescomplemented
20#1556C412yescomplemented
21#1542—22no—
22#1485—22no—
23#1485C414yescomplemented
24C2×Q8⋊C8—21no—
25C2×Q8⋊C8—21no—
26#1455—22no—
27#1453—22no—
28#1452—21no—
29#1450—21no—
30#1298—22no—
31#1288—22no—
32#1286—22no—
33C22×M16—21no—
34#533—21no—
35#533—21no—
36#533—21no—
37#533—21no—
38#533—21no—
39#533—21no—
40#533C2×C412yes—
41#533C2×C412yes—
42C2×C4⋊Q8C2×C411yescharacteristic
43C2×C4⋊Q8C2×C411yescharacteristic
44C2×#125—21no—
45C2×C22⋊Q8—21no—
46C2×C22⋊Q8—21no—
47C2×C4×Q8—21no—
48C2×C4×Q8—21no—
49C2×#117—21no—
50C22×C42C2311yescharacteristic
51#432—21no—
52#432—21no—
53#432—21no—
54#432—21no—
55#432—21no—
56#432—21no—
57#432C2×C412yes—
58#432C2×C412yes—
59#431—21no—
60#431—21no—
61#431—21no—
62#431—21no—
63#431—21no—
64#431—21no—
65#431C2×C412yes—
66#431D412yes—
67#423C2×C412yes—
68#423D412yes—
69C2×C4⋊C8—21no—
70C2×C4⋊C8—21no—
71#407—21no—
72#407D412yes—
73C2×C22⋊C8—21no—
74C2×C22⋊C8—21no—
75C4×M16—21no—
76C4×M16—21no—
77C4×M16—21no—
78C4×M16C2×C412yes—
79C4×M16D412yes—
80C4×M16D412yes—
81C2×#97—21no—
82C2×#97—21no—
83C2×C4×C8—21no—
84C2×C4×C8D411yescharacteristic
85C2×C4×C8D411yescharacteristic
86#383—21no—
87#382—21no—
88#381—21no—
89#380—21no—
90C4×C4⋊C4—21no—
91C4×C22⋊C4—21no—
92#376C2×C412yes—
93#339—28no—
94#333—24no—
95#332—24no—
96Q8⋊C8—24no—
97Q8⋊C8—24no—
98#322—22no—
99#322—24no—
100C82—22no—
101C22×Q8—21no—
102C22×Q8—21no—
103C23×C4C2×D411yescharacteristic
104C23×C4#4111yescharacteristic
105C23×C4#4111yescharacteristic
106C2×M16—22no—
107C2×M16—22no—
108C2×M16—22no—
109C2×M16—44no—
110C2×M16C22⋊C414yes—
111C22×C8—21no—
112C22×C8—21no—
113C22×C8—41no—
114C22×C8—41no—
115C4⋊Q8—21no—
116C4⋊Q8—21no—
117C4⋊Q8—21no—
118C4⋊Q8—21no—
119C4⋊Q8—21no—
120C4⋊Q8—21no—
121C4⋊Q8C4212yes—
122C4⋊Q8C4212yes—
123#125—22no—
124#125—22no—
125#125—41no—
126#125—41no—
127C22⋊Q8—22no—
128C22⋊Q8—22no—
129C22⋊Q8—22no—
130C22⋊Q8—22no—
131C22⋊Q8—41no—
132C22⋊Q8—41no—
133C22⋊Q8—41no—
134C22⋊Q8—41no—
135C4×Q8—22no—
136C4×Q8—22no—
137C4×Q8—22no—
138C4×Q8—22no—
139C4×Q8—41no—
140C4×Q8—41no—
141C4×Q8—41no—
142C4×Q8—41no—
143#117—22no—
144#117—22no—
145#117—41no—
146#117—41no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—41no—
152C2×C4⋊C4—41no—
153C2×C4⋊C4—41no—
154C2×C22⋊C4—41no—
155C2×C22⋊C4—41no—
156C2×C42—22no—
157C2×C42—41no—
158C2×C42C22×C411yescharacteristic
159C2×C42C22×C411yescharacteristic
160C2×C42C22×C411yescharacteristic
161C2×C42C2×D411yescharacteristic
162C2×C42C2×D411yescharacteristic
163C2×C42C2×D411yescharacteristic
164C2×C42C2×D411yescharacteristic
165C2×C42C2×D411yescharacteristic
166C2×C42#4111yescharacteristic
167C2×C42#4111yescharacteristic
168C2×C42#4111yescharacteristic
169C2×C42#4111yescharacteristic
170C4⋊C8—22no—
171C4⋊C8—22no—
172C4⋊C8—22no—
173C4⋊C8—22no—
174C4⋊C8—22no—
175C4⋊C8—41no—
176C4⋊C8—41no—
177C4⋊C8C22⋊C414yes—
178C22⋊C8—22no—
179C22⋊C8—22no—
180C22⋊C8—22no—
181C22⋊C8—22no—
182C22⋊C8—22no—
183C22⋊C8—22no—
184C22⋊C8—41no—
185C22⋊C8—41no—
186#97—22no—
187#97—22no—
188#97—22no—
189#97—22no—
190#97—22no—
191#97—41no—
192C4×C8—22no—
193C4×C8—21no—
194C4×C8—22no—
195C4×C8—22no—
196C4×C8—21no—
197C4×C8—41no—
198C4×C8C22⋊C412yes—
199C4×C8C22⋊C412yes—
200#95—22no—
201#95—22no—
202#95—41no—
203#95—41no—
204C24#12411yescharacteristic
205C2×Q8—22no—
206C2×Q8—22no—
207C2×Q8—44no—
208C2×Q8—41no—
209C2×Q8—41no—
210C2×Q8—82no—
211C22×C4—42no—
212C22×C4—42no—
213C22×C4—42no—
214C22×C4—41no—
215C22×C4—41no—
216C22×C4—42no—
217C22×C4C2×C22⋊C411yescharacteristic
218C22×C4#11711yescharacteristic
219C22×C4#11711yescharacteristic
220C22×C4C4×D411yescharacteristic
221C22×C4C4×D411yescharacteristic
222C22×C4C4×D411yescharacteristic
223C22×C4C4×D411yescharacteristic
224C22×C4C4×D411yescharacteristic
225C22×C4C4×D411yescharacteristic
226C22×C4#12111yescharacteristic
227C22×C4#12111yescharacteristic
228C22×C4#12111yescharacteristic
229C22×C4#12111yescharacteristic
230C22×C4#12311yescharacteristic
231C22×C4#12311yescharacteristic
232C22×C4#12611yescharacteristic
233C22×C4#12611yescharacteristic
234C22×C4#12711yescharacteristic
235M16—44no—
236M16—44no—
237M16—44no—
238M16—82no—
239C2×C8—24no—
240C2×C8—22no—
241C2×C8—22no—
242C2×C8—24no—
243C2×C8—24no—
244C2×C8—42no—
245C2×C8—42no—
246C2×C8—42no—
247C2×C8—82no—
248C2×C8—82no—
249C4⋊C4—22no—
250C4⋊C4—22no—
251C4⋊C4—22no—
252C4⋊C4—22no—
253C4⋊C4—41no—
254C4⋊C4—42no—
255C4⋊C4—41no—
256C4⋊C4—42no—
257C4⋊C4—42no—
258C4⋊C4—42no—
259C4⋊C4—41no—
260C4⋊C4—41no—
261C4⋊C4—82no—
262C22⋊C4—42no—
263C22⋊C4—42no—
264C22⋊C4—82no—
265C42—44no—
266C42—42no—
267C42—42no—
268C42C2×C4211yescharacteristic
269C42C2×C22⋊C411yescharacteristic
270C42C2×C22⋊C411yescharacteristic
271C42#11711yescharacteristic
272C42C4×D411yescharacteristic
273C42C4×D411yescharacteristic
274C42C4×D411yescharacteristic
275C42C4×D411yescharacteristic
276C42C4×D411yescharacteristic
277C42C4×D411yescharacteristic
278C42#12011yescharacteristic
279C42#12111yescharacteristic
280C42#12111yescharacteristic
281C42#12311yescharacteristic
282C42#12411yescharacteristic
283C42#12411yescharacteristic
284C23—22no—
285C23—41no—
286C23—41no—
287C23#38811yescharacteristic
288C23#38811yescharacteristic
289C23#39011yescharacteristic
290Q8—44no—
291Q8—82no—
292Q8—82no—
293Q8—82no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—22no—
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—41no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—82no—
315C2×C4—82no—
316C2×C4—82no—
317C2×C4—82no—
318C2×C4—82no—
319C2×C4C4×C22⋊C411yescharacteristic
320C2×C4C4×C22⋊C411yescharacteristic
321C2×C4C4×C22⋊C411yescharacteristic
322C2×C4#38611yescharacteristic
323C2×C4#38611yescharacteristic
324C2×C4#38611yescharacteristic
325C2×C4#38811yescharacteristic
326C2×C4#38811yescharacteristic
327C2×C4#38811yescharacteristic
328C2×C4#38811yescharacteristic
329C2×C4#39011yescharacteristic
330C2×C4#39011yescharacteristic
331C8—24no—
332C8—42no—
333C8—44no—
334C8—44no—
335C8—82no—
336C8—82no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—82no—
346C22#143711yescharacteristic
347C22#193212yes—
348C4—21no—
349C4—21no—
350C4—21no—
351C4—21no—
352C4—21no—
353C4—21no—
354C4—21no—
355C4—21no—
356C4—21no—
357C4—21no—
358C4—82no—
359C4—82no—
360C4—162no—
361C4#188112yes—
362C4#190912yes—
363C2—22no—
364C2—41no—
365C2—41no—
366C2#1094611yescharacteristic
367C2#1094711yescharacteristic
368C2#1095611yescharacteristic
subgroupG/H
centerC2×C4#386
commutatorC42C2×C42
Frattini, cosocleC22×C42C23
socleC22#1437
prime radicalC24#124
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
19. #1558C4semidirect
20. #1556C4semidirect
23. #1485C4semidirect

Maximal subgroups

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

subgroupindex
#218452
#177342
C2×#14852
#173342
#80582
#79722
#79562