Finite Group Databasebeta

← #24919359

Subgroups — #24919359

Order 10624.

398 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#19128360C211yesmaximal, characteristic, complemented
2#19128058C211yesmaximal, characteristic, complemented
3#19128042C211yesmaximal, characteristic, complemented
4#19128019C211yesmaximal, characteristic, complemented
5#19128005C211yesmaximal, characteristic, complemented
6#19127935C211yesmaximal, characteristic, complemented
7#19127928C211yesmaximal, characteristic, complemented
8#19127916C211yesmaximal, characteristic, complemented
9#19127844C211yesmaximal, characteristic, complemented
10#19127839C211yesmaximal, characteristic, complemented
11C4×#13050578C211yesmaximal, characteristic, complemented
12D4×C83⋊C8C211yesmaximal, characteristic, complemented
13#19127695C211yesmaximal, characteristic, complemented
14C2×#14581715C211yesmaximal, characteristic, complemented
15#553×C83C211yesmaximal, characteristic, complemented
16C2×#13050590C2211yescharacteristic
17C22×Dic166C2211yescharacteristic
18C2×#13050578C2211yescharacteristic
19C2×#13050578C2211yescharacteristic
20C2×#13050575C2211yescharacteristic
21C22×C83⋊C8C2211yescharacteristic
22C22×C83⋊C8C2211yescharacteristic
23#14581780C2211yescharacteristic
24#14581778—21no—
25#14581771C2211yescharacteristic
26#14581770—21no—
27#14581766C2211yescharacteristic
28#14581764—21no—
29C4×Dic166C2211yescharacteristic
30(C22⋊D83)⋊C4—21no—
31#14581725C2211yescharacteristic
32#14581723C2211yescharacteristic
33#14581723C2211yescharacteristic
34C4⋊(C83⋊C8)C2211yescharacteristic
35#14581715C2211yescharacteristic
36#14581715C2211yescharacteristic
37#14581715C2211yescharacteristic
38#14581715C2211yescharacteristic
39#14581715C2211yescharacteristic
40#14581715C2211yescharacteristic
41#14581713C2211yescharacteristic
42#14581712C2211yescharacteristic
43#14581712C2211yescharacteristic
44#14581712C2211yescharacteristic
45C22⋊(C83⋊C8)C2211yescharacteristic
46C22⋊(C83⋊C8)C2211yescharacteristic
47C4×C83⋊C8C2211yescharacteristic
48#126×C83—21no—
49#125×C83C2211yescharacteristic
50#123×C83—21no—
51#121×C83—21no—
52#121×C83C2211yescharacteristic
53#121×C83C2211yescharacteristic
54C4×D4×C83—21no—
55C4×D4×C83C2211yescharacteristic
56C4×D4×C83C2211yescharacteristic
57#117×C83C2211yescharacteristic
58C2×C4⋊C4×C83C2211yescharacteristic
59#13050590—22no—
60#13050590—41no—
61C2×Dic166—22no—
62C2×Dic166—21no—
63C2×Dic166—41no—
64C2×Dic166C2311yescharacteristic
65C2×Dic166C2311yescharacteristic
66C2×D332C2311yescharacteristic
67C2×C22⋊D83—21no—
68C2×C22⋊D83—21no—
69C22×Dic83—21no—
70C2×C4×D83—21no—
71#13050578—22no—
72#13050578—41no—
73#13050575—41no—
74C2×C83⋊C8—41no—
75C2×C83⋊C8—41no—
76C2×C83⋊C8C2311yescharacteristic
77C2×C83⋊C8C2311yescharacteristic
78C2×C83⋊C8C2311yescharacteristic
79C2×C83⋊C8C2311yescharacteristic
80C4⋊Dic83C2311yescharacteristic
81Dic83⋊C4—21no—
82Dic83⋊C4—21no—
83C22⋊Dic83—21no—
84(C2×D83)⋊C4—21no—
85(C2×D83)⋊C4—21no—
86(C2×D83)⋊C4—21no—
87C4×Dic83—21no—
88C2×D4×C83—21no—
89C2×D4×C83—21no—
90C2×D4×C83D411yescharacteristic
91C2×D4×C83C2311yescharacteristic
92C22×C4×C83—21no—
93C22×C4×C83—21no—
94C22×C4×C83C2311yescharacteristic
95C22×C4×C83C2311yescharacteristic
96C4⋊C4×C83—21no—
97C4⋊C4×C83—21no—
98C4⋊C4×C83—21no—
99C4⋊C4×C83D411yescharacteristic
100C4⋊C4×C83C2311yescharacteristic
101C4⋊C4×C83C2311yescharacteristic
102C4⋊C4×C83C2311yescharacteristic
103C22⋊C4×C83—21no—
104C22⋊C4×C83—21no—
105C22⋊C4×C83—21no—
106C22⋊C4×C83—21no—
107C22⋊C4×C83D411yescharacteristic
108C22⋊C4×C83D411yescharacteristic
109C42×C83—21no—
110C42×C83C2311yescharacteristic
111C23×C83—21no—
112C23×C83C2×D411yescharacteristic
113C23×C83C2×D411yescharacteristic
114C22×D83—21no—
115D4×C83—22no—
116D4×C83—21no—
117D4×C83—21no—
118D4×C83—42no—
119D4×C83—81no—
120C2×C4×C83—21no—
121C2×C4×C83—21no—
122C2×C4×C83—21no—
123C2×C4×C83—21no—
124C2×C4×C83—21no—
125C2×C4×C83—21no—
126C2×C4×C83—21no—
127C2×C4×C83—81no—
128C2×C4×C83—81no—
129C2×C4×C83C2×D411yescharacteristic
130C2×C4×C83C2×D411yescharacteristic
131C2×C4×C83C2×D411yescharacteristic
132C2×C4×C83C2×D411yescharacteristic
133C2×C4×C83C2411yescharacteristic
134C22⋊D83—42no—
135C22⋊D83—42no—
136C2×Dic83—21no—
137C2×Dic83—21no—
138C2×Dic83—21no—
139C2×Dic83—42no—
140D332—21no—
141D332—21no—
142C4×D83—42no—
143Dic166—21no—
144Dic166—21no—
145Dic166—21no—
146Dic166—42no—
147Dic166#4112yes—
148C83⋊C8—21no—
149C83⋊C8—21no—
150C83⋊C8—21no—
151C83⋊C8—21no—
152C22×C83—21no—
153C22×C83—21no—
154C22×C83—21no—
155C22×C83—21no—
156C22×C83—21no—
157C22×C83—41no—
158C22×C83—41no—
159C22×C83—41no—
160C22×C83#13512yes—
161C22×C83C22×D411yescharacteristic
162C2×D83—41no—
163C2×D83—41no—
164C2×D83—41no—
165C4×C83—21no—
166C4×C83—21no—
167C4×C83—21no—
168C4×C83—21no—
169C4×C83—41no—
170C4×C83—41no—
171C4×C83—41no—
172C4×C83C2×#4111yescharacteristic
173C4×C83H(5,𝔽2)11yescharacteristic
174Dic83—22no—
175Dic83—41no—
176Dic83—41no—
177C2×C83—22no—
178C2×C83—41no—
179C2×C83—81no—
180C2×C83#54611yescharacteristic
181C2×C83C2×#13511yescharacteristic
182C2×C83#57711yescharacteristic
183D83—81no—
184#3301—831nomaximal, Sylow
185C83#330111yescharacteristic, complemented, Sylow
186#553D8311yescharacteristic, complemented
187#548—831no—
188#488—831no—
189#485—831no—
190#481—831no—
191#479—831no—
192#466—831no—
193#465—831no—
194#463—831no—
195#449—831no—
196#448—831no—
197C4×SD16—831no—
198C8×D4—831no—
199#416—831no—
200C2×#103—831no—
201C2×#41—831no—
202C22×Q8—831no—
203C2×SD16—831no—
204C2×SD16—831no—
205C2×D8—831no—
206C22×C8—831no—
207C22×C8—831no—
208#126—21no—
209#125C2×D8311yescharacteristic
210#124—831no—
211#124—1661no—
212#123—21no—
213C22⋊Q8—831no—
214C22⋊Q8—1661no—
215#121—21no—
216#121—831no—
217#121—1661no—
218#121C2×D8311yescharacteristic
219#121C2×D8311yescharacteristic
220C4×Q8—831no—
221C4×D4—21no—
222C4×D4—1661no—
223C4×D4C2×D8311yescharacteristic
224C4×D4C2×D8311yescharacteristic
225#117C2×D8311yescharacteristic
226C2×C4⋊C4C2×D8311yescharacteristic
227#107—831no—
228#106—831no—
229#106—831no—
230C4⋊C8—831no—
231#103—831no—
232#103—831no—
233#103—831no—
234#103—831no—
235#103—831no—
236#103—831no—
237#102—831no—
238#102—831no—
239#102—831no—
240#102—831no—
241C22⋊C8—831no—
242C22⋊C8—831no—
243C4×C8—831no—
244#41—3321no—
245#41—1662no—
246C2×Q8—3321no—
247C2×Q8—831no—
248C2×Q8—831no—
249C2×Q8—1662no—
250C2×Q8—1661no—
251C2×D4—21no—
252C2×D4—21no—
253C2×D4—831no—
254C2×D4—1661no—
255C2×D4—1661no—
256C2×D4C22⋊D8311yescharacteristic
257C2×D4C22×D8311yescharacteristic
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—1661no—
261C22×C4—1661no—
262C22×C4C22×D8311yescharacteristic
263C22×C4C22×D8311yescharacteristic
264SD16—3321no—
265SD16—1662no—
266D8—3321no—
267C2×C8—3321no—
268C2×C8—3321no—
269C2×C8—831no—
270C2×C8—831no—
271C2×C8—831no—
272C2×C8—831no—
273C4⋊C4—21no—
274C4⋊C4—21no—
275C4⋊C4—21no—
276C4⋊C4—831no—
277C4⋊C4—1661no—
278C4⋊C4—1661no—
279C4⋊C4C22⋊D8311yescharacteristic
280C4⋊C4C22×D8311yescharacteristic
281C4⋊C4C22×D8311yescharacteristic
282C4⋊C4C22×D8311yescharacteristic
283C22⋊C4—21no—
284C22⋊C4—21no—
285C22⋊C4—21no—
286C22⋊C4—21no—
287C22⋊C4—1661no—
288C22⋊C4—1661no—
289C22⋊C4—1661no—
290C22⋊C4—1661no—
291C22⋊C4C22⋊D8311yescharacteristic
292C22⋊C4C22⋊D8311yescharacteristic
293C42—21no—
294C42—1661no—
295C42C22×D8311yescharacteristic
296C23—21no—
297C23—1661no—
298C23C2×C22⋊D8311yescharacteristic
299C23C2×C22⋊D8311yescharacteristic
300Q8—3322no—
301Q8—832no—
302Q8—1661no—
303Q8—1661no—
304Q8—1661no—
305D4—22no—
306D4—21no—
307D4—21no—
308D4—42no—
309D4—81no—
310D4—3322no—
311D4—3322no—
312D4—1661no—
313D4—1661no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—81no—
322C2×C4—81no—
323C2×C4—3322no—
324C2×C4—3322no—
325C2×C4—1661no—
326C2×C4—1661no—
327C2×C4—1661no—
328C2×C4C2×C22⋊D8311yescharacteristic
329C2×C4C2×C22⋊D8311yescharacteristic
330C2×C4C2×C22⋊D8311yescharacteristic
331C2×C4C2×C22⋊D8311yescharacteristic
332C2×C4C23×D8311yescharacteristic
333C8—1661no—
334C8—1661no—
335C8—1661no—
336C8—1661no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—3321no—
346C22—3321no—
347C22—3321no—
348C22#1458181212yes—
349C22C22×C22⋊D8311yescharacteristic
350C4—21no—
351C4—21no—
352C4—21no—
353C4—21no—
354C4—41no—
355C4—41no—
356C4—41no—
357C4—3321no—
358C4—3321no—
359C4—1662no—
360C4#41×D8311yescharacteristic
361C4#1458183811yescharacteristic
362C2—22no—
363C2—41no—
364C2—81no—
365C2—6641no—
366C2#1912833911yescharacteristic
367C2C2×#1458181211yescharacteristic
368C2#1912851211yescharacteristic
subgroupG/H
centerC22C22×C22⋊D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#553×C83C2
socleC22×C83C22×D4
hypercenter#553D83
hypocenterC83#3301
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C4×D4 ◁ #553

Complements

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

NH ≅ G/Ntype
1. #19128360C2semidirect
2. #19128058C2semidirect
3. #19128042C2semidirect
4. #19128019C2semidirect
5. #19128005C2semidirect
6. #19127935C2semidirect
7. #19127928C2semidirect
8. #19127916C2semidirect
9. #19127844C2semidirect
10. #19127839C2semidirect
11. C4×#13050578C2semidirect
12. D4×C83⋊C8C2semidirect
13. #19127695C2semidirect
14. C2×#14581715C2semidirect
15. #553×C83C2semidirect
185. C83#3301semidirect
186. #553D83semidirect

Maximal subgroups

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

subgroupindex
#191283602
#191280582
#191280422
#191280192
#191280052
#191279352
#191279282
#191279162
#191278442
#191278392
C4×#130505782
D4×C83⋊C82
#191276952
C2×#145817152
#553×C832
#330183