Finite Group Databasebeta

← #117352

Subgroups — #117352

Order 512.

583 conjugacy classes of subgroups.

The subgroup lattice has 395 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21363C211yesmaximal, characteristic
2#17755C211yesmaximal, characteristic
3#17596C211yesmaximal, characteristic
4C2×#1503C211yesmaximal, characteristic
5#8089C211yesmaximal, characteristic
6C2×#1285C211yesmaximal, characteristic
7#7928C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9C2×#432C2211yescharacteristic
10C2×#432C2211yescharacteristic
11C2×#432C2211yescharacteristic
12C2×#431C2211yescharacteristic
13#2283C2211yescharacteristic
14C2×C4×C4⋊C4C2211yescharacteristic
15#1949—22no—
16#1853—21no—
17#1852—21no—
18#1776—22no—
19#1556C412yescomplemented
20#1556C412yescomplemented
21#1552—22no—
22#1552—22no—
23#1520—21no—
24#1520—21no—
25#1515—22no—
26#1506—22no—
27#1503—22no—
28#1503—22no—
29C2×#328—22no—
30#1302—22no—
31#1285—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×C4⋊C4—21no—
50C22×C42—21no—
51C22×C42C2311yescharacteristic
52#432—21no—
53#432—21no—
54#432—21no—
55#432—21no—
56#432—21no—
57#432—21no—
58#432—21no—
59#432—21no—
60#432C2×C412yes—
61#432C2×C412yes—
62#432D412yes—
63#432D412yes—
64#431—21no—
65#431—21no—
66#431—21no—
67#431—21no—
68C2×#107—22no—
69C2×#106—22no—
70C2×C4⋊C8—21no—
71C2×C4⋊C8—21no—
72C2×C4⋊C8—21no—
73C2×C4⋊C8D412yes—
74C2×C22⋊C8—21no—
75C2×C22⋊C8—21no—
76C2×C22⋊C8—21no—
77C2×C22⋊C8D412yes—
78C2×#97—21no—
79C2×#97—21no—
80C2×#97—21no—
81C2×C4×C8—21no—
82#383—21no—
83#382—21no—
84#381—21no—
85#380D411yescharacteristic
86#380D411yescharacteristic
87C4×C4⋊C4—21no—
88C4×C4⋊C4—21no—
89C4×C4⋊C4—21no—
90C4×C4⋊C4—21no—
91C4×C4⋊C4D412yes—
92C4×C22⋊C4—21no—
93#376—21no—
94C2×#95—21no—
95#340—22no—
96#340—22no—
97#332—24no—
98#332—24no—
99#328—28no—
100#328—24no—
101#328—24no—
102C22×Q8—21no—
103C22×Q8—21no—
104C23×C4—41no—
105C23×C4C2×D411yescharacteristic
106C23×C4C2×D411yescharacteristic
107C23×C4#4111yescharacteristic
108C22×C8—22no—
109C22×C8—22no—
110C22×C8—41no—
111C22×C8—41no—
112C4⋊Q8—21no—
113C4⋊Q8—21no—
114C4⋊Q8—21no—
115C4⋊Q8—21no—
116C4⋊Q8—21no—
117C4⋊Q8—21no—
118C4⋊Q8C4⋊C412yes—
119C4⋊Q8C4⋊C412yes—
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—22no—
141#117—22no—
142#117—41no—
143#117—41no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—44no—
152C2×C4⋊C4—41no—
153C2×C4⋊C4—41no—
154C2×C4⋊C4—41no—
155C2×C22⋊C4—21no—
156C2×C22⋊C4—21no—
157C2×C22⋊C4—41no—
158C2×C42—22no—
159C2×C42—41no—
160C2×C42—41no—
161C2×C42—41no—
162C2×C42—82no—
163C2×C42C22×C411yescharacteristic
164C2×C42C2×D411yescharacteristic
165C2×C42C2×D411yescharacteristic
166C2×C42C2×D411yescharacteristic
167C2×C42C2×D411yescharacteristic
168C2×C42C2×D411yescharacteristic
169C2×C42C2×Q811yescharacteristic
170C2×C42#4111yescharacteristic
171C2×C42#4111yescharacteristic
172C2×C42#4111yescharacteristic
173C2×C42#4111yescharacteristic
174C2×C42#4111yescharacteristic
175#107—44no—
176#106—44no—
177C4⋊C8—22no—
178C4⋊C8—22no—
179C4⋊C8—22no—
180C4⋊C8—41no—
181C4⋊C8—41no—
182C4⋊C8—41no—
183C4⋊C8—41no—
184C4⋊C8C22⋊C414yes—
185C22⋊C8—22no—
186C22⋊C8—22no—
187C22⋊C8—22no—
188C22⋊C8—22no—
189C22⋊C8—22no—
190C22⋊C8—22no—
191C22⋊C8—41no—
192C22⋊C8—41no—
193#97—22no—
194#97—22no—
195#97—22no—
196#97—22no—
197#97—41no—
198#97—41no—
199C4×C8—41no—
200C4×C8—41no—
201#95—22no—
202#95—21no—
203#95—21no—
204#95—41no—
205#95—41no—
206#95—41no—
207#95—41no—
208C24#12111yescharacteristic
209C2×Q8—22no—
210C2×Q8—22no—
211C2×Q8—41no—
212C2×Q8—41no—
213C2×Q8—82no—
214C2×Q8—82no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—42no—
218C22×C4—44no—
219C22×C4—41no—
220C22×C4—41no—
221C22×C4—42no—
222C22×C4—41no—
223C22×C4—41no—
224C22×C4—81no—
225C22×C4—82no—
226C22×C4—82no—
227C22×C4#11711yescharacteristic
228C22×C4C4×D411yescharacteristic
229C22×C4C4×D411yescharacteristic
230C22×C4#12011yescharacteristic
231C22×C4#12011yescharacteristic
232C22×C4#12011yescharacteristic
233C22×C4#12111yescharacteristic
234C22×C4#12111yescharacteristic
235C22×C4#12111yescharacteristic
236C22×C4#12111yescharacteristic
237C22×C4C22⋊Q811yescharacteristic
238C22×C4C22⋊Q811yescharacteristic
239C22×C4C22⋊Q811yescharacteristic
240C22×C4#12311yescharacteristic
241C22×C4#12411yescharacteristic
242C22×C4#12611yescharacteristic
243C22×C4#12611yescharacteristic
244C22×C4#12711yescharacteristic
245C2×C8—24no—
246C2×C8—42no—
247C2×C8—42no—
248C2×C8—42no—
249C2×C8—42no—
250C2×C8—42no—
251C2×C8—82no—
252C2×C8—82no—
253C2×C8—82no—
254C4⋊C4—24no—
255C4⋊C4—22no—
256C4⋊C4—22no—
257C4⋊C4—22no—
258C4⋊C4—22no—
259C4⋊C4—42no—
260C4⋊C4—42no—
261C4⋊C4—42no—
262C4⋊C4—44no—
263C4⋊C4—41no—
264C4⋊C4—41no—
265C4⋊C4—41no—
266C4⋊C4—41no—
267C4⋊C4—42no—
268C4⋊C4—42no—
269C4⋊C4—42no—
270C4⋊C4—42no—
271C22⋊C4—42no—
272C22⋊C4—42no—
273C22⋊C4—42no—
274C22⋊C4—42no—
275C42—42no—
276C42—42no—
277C42—82no—
278C42—82no—
279C42—82no—
280C42C2×C22⋊C411yescharacteristic
281C42C2×C4⋊C411yescharacteristic
282C42C4×D411yescharacteristic
283C42C4×D411yescharacteristic
284C42#12111yescharacteristic
285C42#12111yescharacteristic
286C42#12111yescharacteristic
287C42C22⋊Q811yescharacteristic
288C42C22⋊Q811yescharacteristic
289C42C22⋊Q811yescharacteristic
290C42#12311yescharacteristic
291C42#12311yescharacteristic
292C42#12311yescharacteristic
293C42#12311yescharacteristic
294C42#12411yescharacteristic
295C42#12411yescharacteristic
296C23—41no—
297C23—41no—
298C23#38811yescharacteristic
299C23#39211yescharacteristic
300C23#39611yescharacteristic
301C23#47112yes—
302C23#47312yes—
303Q8—82no—
304Q8—82no—
305Q8—82no—
306Q8—82no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—44no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—82no—
331C2×C4—82no—
332C2×C4—82no—
333C2×C4—88no—
334C2×C4—82no—
335C2×C4—81no—
336C2×C4—81no—
337C2×C4—81no—
338C2×C4—81no—
339C2×C4—81no—
340C2×C4—81no—
341C2×C4#38011yescharacteristic
342C2×C4#38511yescharacteristic
343C2×C4#38511yescharacteristic
344C2×C4#38611yescharacteristic
345C2×C4#38811yescharacteristic
346C2×C4#39011yescharacteristic
347C2×C4#39311yescharacteristic
348C2×C4#39411yescharacteristic
349C2×C4#39411yescharacteristic
350C2×C4#39611yescharacteristic
351C2×C4#39711yescharacteristic
352C2×C4#39911yescharacteristic
353C8—44no—
354C8—82no—
355C8—82no—
356C8—82no—
357C22—21no—
358C22—21no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—21no—
363C22—42no—
364C22—42no—
365C22#143911yescharacteristic
366C22#193012yes—
367C22#201911yescharacteristic
368C22#202011yescharacteristic
369C22#206412yes—
370C4—21no—
371C4—21no—
372C4—21no—
373C4—21no—
374C4—21no—
375C4—21no—
376C4—21no—
377C4—21no—
378C4—21no—
379C4—21no—
380C4—21no—
381C4—84no—
382C4—82no—
383C4—82no—
384C4—161no—
385C4—161no—
386C4#190612yes—
387C2—21no—
388C2—21no—
389C2—22no—
390C2—22no—
391C2#1106011yescharacteristic
392C2#1113211yescharacteristic
393C2#1114511yescharacteristic
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
19. #1556C4semidirect
20. #1556C4semidirect

Maximal subgroups

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

subgroupindex
#213632
#177552
#175962
C2×#15032
#80892
C2×#12852
#79282