Finite Group Databasebeta

← #24914501

Subgroups — #24914501

Order 10624.

552 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#19127637C211yesmaximal, characteristic, complemented
2#19127630C211yesmaximal, characteristic, complemented
3#19127629C211yesmaximal, characteristic, complemented
4#19127627C211yesmaximal, characteristic, complemented
5#19127625C211yesmaximal, characteristic, complemented
6#19127623C211yesmaximal, characteristic, complemented
7#19127595C211yesmaximal, characteristic, complemented
8#19127593C211yesmaximal, characteristic, complemented
9#19127592C211yesmaximal, characteristic, complemented
10#19127592C211yesmaximal, characteristic, complemented
11#19127590C211yesmaximal, characteristic, complemented
12#19127556C211yesmaximal, characteristic
13#19127556C211yesmaximal, characteristic
14#19127556C211yesmaximal, characteristic
15#388×C83C211yesmaximal, characteristic, complemented
16C24×D83C2211yescharacteristic
17C2×C4⋊Dic83C2211yescharacteristic, complemented
18C2×Dic83⋊C4C2211yescharacteristic, complemented
19C2×Dic83⋊C4C2211yescharacteristic, complemented
20C2×Dic83⋊C4C2211yescharacteristic, complemented
21C2×Dic83⋊C4C2211yescharacteristic, complemented
22C2×C22⋊Dic83C2211yescharacteristic
23C2×C22⋊Dic83C2211yescharacteristic
24C2×C22⋊Dic83C2211yescharacteristic
25C2×(C2×D83)⋊C4C2211yescharacteristic
26C2×(C2×D83)⋊C4C2211yescharacteristic
27C2×(C2×D83)⋊C4C2211yescharacteristic
28C2×(C2×D83)⋊C4C2211yescharacteristic
29C2×(C2×D83)⋊C4C2211yescharacteristic
30C2×(C2×D83)⋊C4C2211yescharacteristic
31C2×(C2×D83)⋊C4C2211yescharacteristic
32C2×(C2×D83)⋊C4C2211yescharacteristic
33C2×(C2×D83)⋊C4C2211yescharacteristic
34C2×(C2×D83)⋊C4C2211yescharacteristic
35C2×(C2×D83)⋊C4C2211yescharacteristic
36C2×(C2×D83)⋊C4C2211yescharacteristic
37C2×C4×Dic83C2211yescharacteristic, complemented
38C83⋊#95C2211yescharacteristic, complemented
39C83⋊#95C2211yescharacteristic, complemented
40C83⋊#95C2211yescharacteristic, complemented
41C83⋊#95C2211yescharacteristic, complemented
42C83⋊#95C2211yescharacteristic, complemented
43C83⋊#95C2211yescharacteristic, complemented
44C2×C4⋊C4×C83C2211yescharacteristic, complemented
45C2×C22⋊C4×C83C2211yescharacteristic
46C2×C22⋊C4×C83C2211yescharacteristic
47C2×C22⋊C4×C83C2211yescharacteristic
48C2×C42×C83C2211yescharacteristic, complemented
49#95×C83C2211yescharacteristic, complemented
50#95×C83C2211yescharacteristic, complemented
51C23×D83—22no—
52C23×D83—22no—
53C23×D83—22no—
54C23×D83—41no—
55C23×D83—41no—
56C23×D83—41no—
57C23×D83—41no—
58C23×D83C2311yescharacteristic
59C23×D83C2311yescharacteristic
60C22×Dic83C2311yescharacteristic
61C22×Dic83C2311yescharacteristic
62C22×Dic83C2311yescharacteristic
63C22×Dic83C2311yescharacteristic
64C22×Dic83C2311yescharacteristic
65C22×Dic83C2311yescharacteristic
66C4⋊Dic83—41no—
67Dic83⋊C4—41no—
68Dic83⋊C4—41no—
69Dic83⋊C4—41no—
70Dic83⋊C4—41no—
71C22⋊Dic83—22no—
72C22⋊Dic83—41no—
73C22⋊Dic83—41no—
74(C2×D83)⋊C4—22no—
75(C2×D83)⋊C4—22no—
76(C2×D83)⋊C4—22no—
77(C2×D83)⋊C4—22no—
78(C2×D83)⋊C4—41no—
79(C2×D83)⋊C4—41no—
80(C2×D83)⋊C4—41no—
81(C2×D83)⋊C4—41no—
82(C2×D83)⋊C4—41no—
83(C2×D83)⋊C4—41no—
84(C2×D83)⋊C4—41no—
85(C2×D83)⋊C4—41no—
86C4×Dic83—41no—
87C24×C83C2311yescharacteristic
88C22×C4×C83C2311yescharacteristic
89C22×C4×C83C2311yescharacteristic
90C22×C4×C83C2311yescharacteristic
91C22×C4×C83C2311yescharacteristic
92C22×C4×C83C2311yescharacteristic
93C22×C4×C83C2311yescharacteristic
94C4⋊C4×C83—41no—
95C22⋊C4×C83—22no—
96C22⋊C4×C83—41no—
97C22⋊C4×C83—41no—
98C42×C83—41no—
99C23×C83—21no—
100C23×C83—21no—
101C23×C83—21no—
102C23×C83—21no—
103C23×C83—21no—
104C23×C83—21no—
105C23×C83#4112yes—
106C23×C83C2411yescharacteristic
107C22×D83—21no—
108C22×D83—21no—
109C22×D83—21no—
110C22×D83—21no—
111C22×D83—21no—
112C22×D83—21no—
113C22×D83—21no—
114C22×D83—21no—
115C22×D83—21no—
116C22×D83—21no—
117C22×D83—21no—
118C22×D83—21no—
119C22×D83—44no—
120C22×D83—44no—
121C22×D83—44no—
122C22×D83—44no—
123C22×D83—44no—
124C22×D83—44no—
125C22×D83—44no—
126C22×D83#4112yes—
127C22×D83#4112yes—
128C2×C4×C83—21no—
129C2×C4×C83—21no—
130C2×C4×C83—21no—
131C2×C4×C83—21no—
132C2×C4×C83—21no—
133C2×C4×C83—21no—
134C2×C4×C83—21no—
135C2×C4×C83—21no—
136C2×C4×C83—21no—
137C2×C4×C83—21no—
138C2×C4×C83—21no—
139C2×C4×C83—21no—
140C2×C4×C83—21no—
141C2×C4×C83—21no—
142C2×C4×C83—21no—
143C2×C4×C83—21no—
144C2×C4×C83—21no—
145C2×C4×C83—21no—
146C2×Dic83—21no—
147C2×Dic83—21no—
148C2×Dic83—21no—
149C2×Dic83—21no—
150C2×Dic83—21no—
151C2×Dic83—21no—
152C2×Dic83—21no—
153C2×Dic83—21no—
154C2×Dic83—21no—
155C2×Dic83—21no—
156C2×Dic83—21no—
157C2×Dic83—21no—
158C2×Dic83—21no—
159C2×Dic83—21no—
160C2×Dic83—21no—
161C2×Dic83—21no—
162C2×Dic83—21no—
163C2×Dic83—21no—
164C22×C83—22no—
165C22×C83—22no—
166C22×C83—22no—
167C22×C83—41no—
168C22×C83—41no—
169C22×C83—41no—
170C22×C83—41no—
171C22×C83C2×#4111yescharacteristic
172C22×C83C2×#4111yescharacteristic
173C22×C83C2×#4111yescharacteristic
174C22×C83H(5,𝔽2)11yescharacteristic
175C22×C83H(5,𝔽2)11yescharacteristic
176C22×C83H(5,𝔽2)11yescharacteristic
177C22×C83H(5,𝔽2)11yescharacteristic
178C2×D83—22no—
179C2×D83—22no—
180C2×D83—22no—
181C2×D83—22no—
182C2×D83—22no—
183C2×D83—22no—
184C2×D83—416no—
185C2×D83—41no—
186C2×D83—41no—
187C2×D83—41no—
188C2×D83—41no—
189C2×D83—41no—
190C2×D83—41no—
191C2×D83—41no—
192C2×D83—41no—
193C4×C83—41no—
194C4×C83—41no—
195C4×C83—41no—
196C4×C83—41no—
197C4×C83—41no—
198C4×C83—41no—
199Dic83—41no—
200Dic83—41no—
201Dic83—41no—
202Dic83—41no—
203Dic83—41no—
204Dic83—41no—
205C2×C83—42no—
206C2×C83#53811yescharacteristic
207C2×C83#53811yescharacteristic
208C2×C83#53811yescharacteristic
209C2×C83#55111yescharacteristic
210C2×C83#55111yescharacteristic
211C2×C83#55111yescharacteristic
212C2×C83#56011yescharacteristic
213D83—42no—
214D83—42no—
215#2735—831nomaximal, Sylow
216C83#273511yescharacteristic, complemented, Sylow
217#399—831no—
218#397—831no—
219#397—831no—
220#397—831no—
221#396—831no—
222#396—831no—
223#388—831no—
224#388—831no—
225#388—831no—
226#388—831no—
227#388—831no—
228#388D8311yescharacteristic, complemented
229#379—831no—
230#379—831no—
231#379—831no—
232C25—831no—
233C2×C4⋊C4—831no—
234C2×C4⋊C4—831no—
235C2×C4⋊C4—831no—
236C2×C4⋊C4—831no—
237C2×C4⋊C4—831no—
238C2×C4⋊C4C2×D8311yescharacteristic, complemented
239C2×C22⋊C4—831no—
240C2×C22⋊C4—831no—
241C2×C22⋊C4—831no—
242C2×C22⋊C4—831no—
243C2×C22⋊C4—831no—
244C2×C22⋊C4—831no—
245C2×C22⋊C4—831no—
246C2×C22⋊C4—831no—
247C2×C22⋊C4—831no—
248C2×C22⋊C4—831no—
249C2×C22⋊C4—831no—
250C2×C22⋊C4—831no—
251C2×C22⋊C4—831no—
252C2×C22⋊C4—831no—
253C2×C22⋊C4—831no—
254C2×C22⋊C4C2×D8311yescharacteristic
255C2×C22⋊C4C2×D8311yescharacteristic
256C2×C22⋊C4C2×D8311yescharacteristic
257C2×C42—831no—
258C2×C42C2×D8311yescharacteristic, complemented
259#95—831no—
260#95—831no—
261#95—831no—
262#95—831no—
263#95—831no—
264#95—831no—
265#95C2×D8311yescharacteristic, complemented
266#95C2×D8311yescharacteristic, complemented
267C24—3321no—
268C24—3321no—
269C24—3321no—
270C24—3321no—
271C24—831no—
272C24—831no—
273C24—1662no—
274C24—1662no—
275C24—1662no—
276C24C22×D8311yescharacteristic
277C22×C4—831no—
278C22×C4—831no—
279C22×C4—831no—
280C22×C4—831no—
281C22×C4—831no—
282C22×C4—831no—
283C22×C4C22×D8311yescharacteristic
284C22×C4C22×D8311yescharacteristic
285C22×C4C22×D8311yescharacteristic
286C22×C4C22×D8311yescharacteristic
287C22×C4C22×D8311yescharacteristic
288C22×C4C22×D8311yescharacteristic
289C4⋊C4—41no—
290C4⋊C4—3321no—
291C4⋊C4—3321no—
292C4⋊C4—3321no—
293C4⋊C4—3321no—
294C4⋊C4—3321no—
295C22⋊C4—22no—
296C22⋊C4—41no—
297C22⋊C4—41no—
298C22⋊C4—3321no—
299C22⋊C4—3321no—
300C22⋊C4—3321no—
301C22⋊C4—3321no—
302C22⋊C4—3321no—
303C22⋊C4—3321no—
304C22⋊C4—3321no—
305C22⋊C4—3321no—
306C22⋊C4—3321no—
307C22⋊C4—3321no—
308C22⋊C4—1662no—
309C22⋊C4—1662no—
310C22⋊C4—1662no—
311C22⋊C4—1662no—
312C22⋊C4—1662no—
313C42—41no—
314C42—3321no—
315C23—21no—
316C23—21no—
317C23—21no—
318C23—21no—
319C23—21no—
320C23—21no—
321C23—3324no—
322C23—3324no—
323C23—3324no—
324C23—3324no—
325C23—3324no—
326C23—3324no—
327C23—3324no—
328C23—832no—
329C23—832no—
330C23—1661no—
331C23—1661no—
332C23—1661no—
333C23—1661no—
334C23—1661no—
335C23—1661no—
336C23—1661no—
337C23—1661no—
338C23—1661no—
339C23—1661no—
340C23—1661no—
341C23—1661no—
342C23#1305059012yes—
343C23C23×D8311yescharacteristic
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—1661no—
363C2×C4—1661no—
364C2×C4—1661no—
365C2×C4—1661no—
366C2×C4—1661no—
367C2×C4—1661no—
368C2×C4—1661no—
369C2×C4—1661no—
370C2×C4—1661no—
371C2×C4—1661no—
372C2×C4—1661no—
373C2×C4—1661no—
374C2×C4—1661no—
375C2×C4—1661no—
376C2×C4—1661no—
377C2×C4—1661no—
378C2×C4—1661no—
379C2×C4—1661no—
380C22—22no—
381C22—22no—
382C22—22no—
383C22—41no—
384C22—41no—
385C22—41no—
386C22—41no—
387C22—33216no—
388C22—3321no—
389C22—3321no—
390C22—3321no—
391C22—3321no—
392C22—3321no—
393C22—3321no—
394C22—3321no—
395C22—3321no—
396C22—1662no—
397C22—1662no—
398C22—1662no—
399C22—1662no—
400C22—1662no—
401C22—1662no—
402C22#41×D8311yescharacteristic
403C22#41×D8311yescharacteristic
404C22C2×#1305059011yescharacteristic
405C22#1458183711yescharacteristic
406C22#1458183711yescharacteristic
407C22#1458183811yescharacteristic
408C22#1458183811yescharacteristic
409C4—41no—
410C4—41no—
411C4—41no—
412C4—41no—
413C4—41no—
414C4—41no—
415C4—3321no—
416C4—3321no—
417C4—3321no—
418C4—3321no—
419C4—3321no—
420C4—3321no—
421C2—42no—
422C2—3322no—
423C2—3322no—
424C2#1912828311yescharacteristic
425C2#1912828411yescharacteristic
426C2#1912828511yescharacteristic
427C2#1912837511yescharacteristic
428C2#1912837611yescharacteristic
429C2#1912838011yescharacteristic
430C2#1912844111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
Fitting#388×C83C2
prime radicalC24×D83C22
hypercenter#388D83
hypocenterC83#2735
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
1. #19127637C2semidirect
2. #19127630C2semidirect
3. #19127629C2semidirect
4. #19127627C2semidirect
5. #19127625C2semidirect
6. #19127623C2semidirect
7. #19127595C2semidirect
8. #19127593C2semidirect
9. #19127592C2semidirect
10. #19127592C2semidirect
11. #19127590C2semidirect
15. #388×C83C2semidirect
17. C2×C4⋊Dic83C22semidirect
18. C2×Dic83⋊C4C22semidirect
19. C2×Dic83⋊C4C22semidirect
20. C2×Dic83⋊C4C22semidirect
21. C2×Dic83⋊C4C22semidirect
37. C2×C4×Dic83C22semidirect
38. C83⋊#95C22semidirect
39. C83⋊#95C22semidirect
40. C83⋊#95C22semidirect
41. C83⋊#95C22semidirect
42. C83⋊#95C22semidirect
43. C83⋊#95C22semidirect
44. C2×C4⋊C4×C83C22semidirect
48. C2×C42×C83C22semidirect
49. #95×C83C22semidirect
50. #95×C83C22semidirect
216. C83#2735semidirect
228. #388D83semidirect
238. C2×C4⋊C4C2×D83semidirect
258. C2×C42C2×D83semidirect
265. #95C2×D83semidirect
266. #95C2×D83semidirect

Maximal subgroups

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

subgroupindex
#191276372
#191276302
#191276292
#191276272
#191276252
#191276232
#191275952
#191275932
#191275922
#191275902
#191275562
#388×C832
#273583