Finite Group Databasebeta

← #24918307

Subgroups — #24918307

Order 10624.

418 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#19128214C211yesmaximal, characteristic, complemented
2#19128213C211yesmaximal, characteristic, complemented
3#19128045C211yesmaximal, characteristic, complemented
4#19128028C211yesmaximal, characteristic, complemented
5#19128025C211yesmaximal, characteristic, complemented
6#19127958C211yesmaximal, characteristic, complemented
7#19127955C211yesmaximal, characteristic, complemented
8#19127950C211yesmaximal, characteristic, complemented
9#19127948C211yesmaximal, characteristic, complemented
10#19127866C211yesmaximal, characteristic, complemented
11#19127861C211yesmaximal, characteristic, complemented
12#19127849C211yesmaximal, characteristic, complemented
13#19127848C211yesmaximal, characteristic, complemented
14#19127663C211yesmaximal, characteristic, complemented
15#485×C83C211yesmaximal, characteristic, complemented
16C2×#13050591C2211yescharacteristic
17C2×#13050590C2211yescharacteristic
18C22×C4×D83C2211yescharacteristic
19C2×#13050580C2211yescharacteristic, complemented
20C2×#13050579C2211yescharacteristic, complemented
21C2×#13050577C2211yescharacteristic, complemented
22C2×D664C2211yescharacteristic, complemented
23C2×#13050573C2211yescharacteristic
24C2×#13050573C2211yescharacteristic
25#14581777—21no—
26#14581776—21no—
27#14581772C2211yescharacteristic
28#14581772C2211yescharacteristic
29#14581768C2211yescharacteristic
30#14581767C2211yescharacteristic
31#14581767C2211yescharacteristic
32#14581767C2211yescharacteristic
33#14581766C2211yescharacteristic
34#14581762—21no—
35#14581762—21no—
36C4×D332—21no—
37C4×D332—21no—
38#14581754—21no—
39(C22⋊D83)⋊C4—21no—
40#14581750C2211yescharacteristic
41#14581725C2211yescharacteristic, complemented
42#14581723C2211yescharacteristic, complemented
43#14581717C2211yescharacteristic, complemented
44#14581715C2211yescharacteristic, complemented
45#14581714C2211yescharacteristic, complemented
46#14581712C2211yescharacteristic, complemented
47#14581712C2211yescharacteristic, complemented
48#14581712C2211yescharacteristic, complemented
49C22⋊(C83⋊C8)C2211yescharacteristic
50(C2×D83)⋊C8C2211yescharacteristic
51(C2×D83)⋊C8C2211yescharacteristic
52C22⋊Q8×C83C2211yescharacteristic
53#117×C83C2211yescharacteristic
54#107×C83C2211yescharacteristic, complemented
55#106×C83C2211yescharacteristic, complemented
56#103×C83C2211yescharacteristic, complemented
57#103×C83C2211yescharacteristic, complemented
58C22⋊C8×C83C2211yescharacteristic
59C23×D83D411yescharacteristic
60#13050591—41no—
61#13050591—41no—
62#13050590—41no—
63#13050590—41no—
64C2×Dic166C2311yescharacteristic
65C2×D332—21no—
66C2×D332—21no—
67C2×D332—21no—
68C2×D332—21no—
69C2×D332C2311yescharacteristic
70C2×D332C2311yescharacteristic
71C2×C22⋊D83—21no—
72C2×C22⋊D83—21no—
73C22×Dic83D411yescharacteristic
74C2×C4×D83—21no—
75C2×C4×D83—21no—
76C2×C4×D83—41no—
77C2×C4×D83—41no—
78C2×C4×D83D411yescharacteristic
79C2×C4×D83D411yescharacteristic
80C2×C4×D83C2311yescharacteristic
81C2×C4×D83C2311yescharacteristic
82#13050580—41no—
83#13050579—41no—
84#13050577—41no—
85D664—41no—
86#13050573—41no—
87#13050573—41no—
88C2×C83⋊C8C2311yescharacteristic
89C2×C83⋊C8C2311yescharacteristic
90C4⋊Dic83—21no—
91C4⋊Dic83—21no—
92C4⋊Dic83C2311yescharacteristic
93Dic83⋊C4—21no—
94(C2×D83)⋊C4—21no—
95(C2×D83)⋊C4—21no—
96(C2×D83)⋊C4—21no—
97(C2×D83)⋊C4—21no—
98(C2×D83)⋊C4—21no—
99(C2×D83)⋊C4—21no—
100(C2×D83)⋊C4—21no—
101(C2×D83)⋊C4—21no—
102C4×Dic83—21no—
103C2×Q8×C83C2311yescharacteristic
104C22×C4×C83C2311yescharacteristic
105C2×C8×C83C2311yescharacteristic
106C2×C8×C83C2311yescharacteristic
107C4⋊C4×C83—21no—
108C4⋊C4×C83C2311yescharacteristic
109C4⋊C4×C83C2311yescharacteristic
110C4⋊C4×C83C2311yescharacteristic
111C22⋊C4×C83—21no—
112C22⋊C4×C83—21no—
113C42×C83—21no—
114C23×C83C2×D411yescharacteristic
115C22×D83—21no—
116C22×D83—21no—
117C22×D83—41no—
118C22×D83—41no—
119C22×D83—41no—
120C22×D83C2×D411yescharacteristic
121C22×D83C2×D411yescharacteristic
122Q8×C83—21no—
123Q8×C83—21no—
124C2×C4×C83—21no—
125C2×C4×C83—21no—
126C2×C4×C83—21no—
127C2×C4×C83—21no—
128C2×C4×C83—21no—
129C2×C4×C83—21no—
130C2×C4×C83C2×D411yescharacteristic
131C2×C4×C83C2411yescharacteristic
132C22⋊D83—42no—
133C22⋊D83—42no—
134C2×Dic83—21no—
135C2×Dic83—21no—
136C2×Dic83—41no—
137C2×Dic83C2×D411yescharacteristic
138C2×Dic83C2×D411yescharacteristic
139D332—21no—
140D332—21no—
141D332—21no—
142D332—21no—
143D332—42no—
144D332—42no—
145D332—42no—
146D332—42no—
147C4×D83—21no—
148C4×D83—21no—
149C4×D83—21no—
150C4×D83—21no—
151C4×D83—41no—
152C4×D83—41no—
153C4×D83—42no—
154C4×D83—42no—
155Dic166—21no—
156Dic166—21no—
157C8×C83—21no—
158C8×C83—21no—
159C83⋊C8—21no—
160C83⋊C8—21no—
161C22×C83—21no—
162C22×C83—21no—
163C22×C83—21no—
164C22×C83C22×D411yescharacteristic
165C2×D83—21no—
166C2×D83—21no—
167C2×D83—21no—
168C2×D83—21no—
169C2×D83—21no—
170C2×D83—21no—
171C2×D83—44no—
172C2×D83—41no—
173C2×D83—41no—
174C2×D83—41no—
175C2×D83—41no—
176C2×D83—41no—
177C2×D83—41no—
178C4×C83—21no—
179C4×C83—41no—
180C4×C83—41no—
181C4×C83—41no—
182C4×C83—41no—
183C4×C83H(5,𝔽2)11yescharacteristic
184C4×C83H(5,𝔽2)11yescharacteristic
185Dic83—21no—
186Dic83—21no—
187Dic83—41no—
188Dic83—41no—
189C2×C83—41no—
190C2×C83#53411yescharacteristic
191C2×C83#57511yescharacteristic
192C2×C83#57511yescharacteristic
193D83—41no—
194D83—41no—
195D83—81no—
196D83—81no—
197#3197—831nomaximal, Sylow
198C83#319711yescharacteristic, complemented, Sylow
199#525—831no—
200#525—831no—
201#485—831no—
202#485D8311yescharacteristic, complemented
203#482—831no—
204#482—831no—
205#470—831no—
206#469—831no—
207#468—831no—
208#468—831no—
209#452—831no—
210#452—831no—
211#449—831no—
212#449—831no—
213#407—831no—
214C2×#41—831no—
215C2×#41—831no—
216C23×C4—831no—
217C2×Q16—831no—
218C2×SD16—831no—
219C2×SD16—831no—
220C2×D8—831no—
221C2×M16—831no—
222C2×M16—831no—
223#123—1661no—
224#123—1661no—
225C22⋊Q8—831no—
226C22⋊Q8—831no—
227C22⋊Q8—831no—
228C22⋊Q8C2×D8311yescharacteristic
229#121—831no—
230#121—831no—
231#121—831no—
232#121—831no—
233#120—1661no—
234#120—1661no—
235C4×D4—1661no—
236C4×D4—1661no—
237C4×D4—1661no—
238C4×D4—1661no—
239#117—831no—
240#117C2×D8311yescharacteristic
241#107—831no—
242#107C2×D8311yescharacteristic, complemented
243#106—831no—
244#106C2×D8311yescharacteristic, complemented
245#103—831no—
246#103—831no—
247#103C2×D8311yescharacteristic, complemented
248#103C2×D8311yescharacteristic, complemented
249#102—831no—
250#102—831no—
251#102—831no—
252#102—831no—
253C22⋊C8—831no—
254C22⋊C8—831no—
255C22⋊C8—831no—
256C22⋊C8C2×D8311yescharacteristic
257C24—831no—
258#41—3321no—
259#41—3321no—
260#41—3321no—
261#41—3321no—
262C2×Q8—831no—
263C2×Q8C22×D8311yescharacteristic
264C2×D4—831no—
265C2×D4—831no—
266C2×D4—1661no—
267C2×D4—1661no—
268C2×D4—1661no—
269C2×D4—1661no—
270C2×D4—1661no—
271C2×D4—1661no—
272C22×C4—3321no—
273C22×C4—3321no—
274C22×C4—831no—
275C22×C4—831no—
276C22×C4—831no—
277C22×C4—831no—
278C22×C4—831no—
279C22×C4—1661no—
280C22×C4—1661no—
281C22×C4C22×D8311yescharacteristic
282Q16—3321no—
283SD16—3321no—
284SD16—3321no—
285D8—3321no—
286M16—3321no—
287M16—3321no—
288C2×C8—831no—
289C2×C8—831no—
290C2×C8C22×D8311yescharacteristic
291C2×C8C22×D8311yescharacteristic
292C4⋊C4—21no—
293C4⋊C4—831no—
294C4⋊C4—1661no—
295C4⋊C4—1661no—
296C4⋊C4—1661no—
297C4⋊C4C22×D8311yescharacteristic
298C4⋊C4C22×D8311yescharacteristic
299C4⋊C4C22×D8311yescharacteristic
300C22⋊C4—21no—
301C22⋊C4—21no—
302C22⋊C4—1661no—
303C22⋊C4—1661no—
304C22⋊C4—1661no—
305C22⋊C4—1661no—
306C22⋊C4—1661no—
307C22⋊C4—1661no—
308C22⋊C4—1661no—
309C22⋊C4—1661no—
310C42—21no—
311C42—1661no—
312C23—3321no—
313C23—3321no—
314C23—3321no—
315C23—831no—
316C23—831no—
317C23—1661no—
318C23—1661no—
319C23D4×D8311yescharacteristic
320Q8—21no—
321Q8—21no—
322Q8—1661no—
323Q8—1661no—
324D4—3322no—
325D4—3322no—
326D4—3322no—
327D4—3322no—
328D4—3322no—
329D4—3322no—
330D4—1661no—
331D4—1661no—
332D4—1661no—
333D4—1661no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—3322no—
341C2×C4—3321no—
342C2×C4—3321no—
343C2×C4—3322no—
344C2×C4—3321no—
345C2×C4—831no—
346C2×C4—831no—
347C2×C4—1661no—
348C2×C4—1661no—
349C2×C4—1661no—
350C2×C4—1661no—
351C2×C4—1661no—
352C2×C4—1661no—
353C2×C4D4×D8311yescharacteristic
354C2×C4C23×D8311yescharacteristic
355C8—21no—
356C8—21no—
357C8—1661no—
358C8—1661no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—3324no—
363C22—3321no—
364C22—3321no—
365C22—3321no—
366C22—3321no—
367C22—3321no—
368C22—3321no—
369C22—1661no—
370C22—1661no—
371C22—1661no—
372C22—1661no—
373C22—1661no—
374C22—1661no—
375C22C2×D4×D8311yescharacteristic
376C4—21no—
377C4—41no—
378C4—41no—
379C4—41no—
380C4—41no—
381C4—3321no—
382C4—3321no—
383C4—1661no—
384C4—1661no—
385C4#1458183811yescharacteristic
386C4#1458183811yescharacteristic
387C2—41no—
388C2—3321no—
389C2—3321no—
390C2—6641no—
391C2—6641no—
392C2#1912826211yescharacteristic
393C2#1912849811yescharacteristic
394C2#1912850011yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#485×C83C2
socleC22×C83C22×D4
hypercenter#485D83
hypocenterC83#3197
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ #485

Complements

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

NH ≅ G/Ntype
1. #19128214C2semidirect
2. #19128213C2semidirect
3. #19128045C2semidirect
4. #19128028C2semidirect
5. #19128025C2semidirect
6. #19127958C2semidirect
7. #19127955C2semidirect
8. #19127950C2semidirect
9. #19127948C2semidirect
10. #19127866C2semidirect
11. #19127861C2semidirect
12. #19127849C2semidirect
13. #19127848C2semidirect
14. #19127663C2semidirect
15. #485×C83C2semidirect
19. C2×#13050580C22semidirect
20. C2×#13050579C22semidirect
21. C2×#13050577C22semidirect
22. C2×D664C22semidirect
41. #14581725C22semidirect
42. #14581723C22semidirect
43. #14581717C22semidirect
44. #14581715C22semidirect
45. #14581714C22semidirect
46. #14581712C22semidirect
47. #14581712C22semidirect
48. #14581712C22semidirect
54. #107×C83C22semidirect
55. #106×C83C22semidirect
56. #103×C83C22semidirect
57. #103×C83C22semidirect
198. C83#3197semidirect
202. #485D83semidirect
242. #107C2×D83semidirect
244. #106C2×D83semidirect
247. #103C2×D83semidirect
248. #103C2×D83semidirect

Maximal subgroups

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

subgroupindex
#191282142
#191282132
#191280452
#191280282
#191280252
#191279582
#191279552
#191279502
#191279482
#191278662
#191278612
#191278492
#191278482
#191276632
#485×C832
#319783