Finite Group Databasebeta

← #24918432

Subgroups — #24918432

Order 10624.

410 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#19128238C211yesmaximal, characteristic, complemented
2#124×D83C211yesmaximal, characteristic, complemented
3#19128086C211yesmaximal, characteristic, complemented
4#19128082C211yesmaximal, characteristic, complemented
5#19128057C211yesmaximal, characteristic, complemented
6#19128054C211yesmaximal, characteristic, complemented
7#19127923C211yesmaximal, characteristic, complemented
8#19127917C211yesmaximal, characteristic, complemented
9#19127913C211yesmaximal, characteristic, complemented
10#19127855C211yesmaximal, characteristic, complemented
11#19127850C211yesmaximal, characteristic, complemented
12#19127841C211yesmaximal, characteristic, complemented
13#19127836C211yesmaximal, characteristic
14#19127753C211yesmaximal, characteristic, complemented
15#464×C83C211yesmaximal, characteristic, complemented
16C2×Q8×D83C2211yescharacteristic
17C2×D4×D83C2211yescharacteristic
18C2×#13050580C2211yescharacteristic, complemented
19C2×#13050579C2211yescharacteristic, complemented
20C2×#13050577C2211yescharacteristic
21C2×#13050577C2211yescharacteristic, complemented
22C2×D664C2211yescharacteristic
23C2×#13050575C2211yescharacteristic
24#14581787—21no—
25#14581783C2211yescharacteristic, complemented
26#14581781C2211yescharacteristic
27#14581781C2211yescharacteristic, complemented
28#14581780C2211yescharacteristic
29#14581779C2211yescharacteristic
30#14581778—21no—
31#14581778—21no—
32#14581777—21no—
33#14581768C2211yescharacteristic
34#14581767C2211yescharacteristic
35C4×D332—21no—
36C4×D332C2211yescharacteristic, complemented
37#14581749—21no—
38C22⋊C4×D83—21no—
39C22⋊C4×D83—21no—
40C42×D83C2211yescharacteristic
41C4⋊(C83⋊C8)C2211yescharacteristic, complemented
42#14581717C2211yescharacteristic, complemented
43#14581716C2211yescharacteristic, complemented
44#14581715C2211yescharacteristic, complemented
45#14581714C2211yescharacteristic
46#14581713C2211yescharacteristic
47#14581712C2211yescharacteristic
48(C2×D83)⋊C8C2211yescharacteristic
49(C2×D83)⋊C8C2211yescharacteristic
50C8×Dic83C2211yescharacteristic, complemented
51C8×Dic83C2211yescharacteristic, complemented
52C2×Q16×C83C2211yescharacteristic, complemented
53C2×SD16×C83C2211yescharacteristic
54#124×C83C2211yescharacteristic
55C4×Q8×C83C2211yescharacteristic, complemented
56C4⋊C8×C83C2211yescharacteristic, complemented
57#103×C83C2211yescharacteristic, complemented
58#102×C83C2211yescharacteristic
59C23×D83—21no—
60C2×Dic166—21no—
61C2×Dic166C2311yescharacteristic
62Q8×D83—41no—
63Q8×D83—41no—
64C2×D332—21no—
65C2×D332C2311yescharacteristic
66C2×D332C2311yescharacteristic
67C2×C22⋊D83—21no—
68D4×D83—41no—
69D4×D83—41no—
70C2×C4×D83—21no—
71C2×C4×D83—21no—
72C2×C4×D83D411yescharacteristic
73C2×C4×D83D411yescharacteristic
74C2×C4×D83C2311yescharacteristic
75#13050580—41no—
76#13050579—41no—
77#13050577—41no—
78#13050577—41no—
79D664—41no—
80#13050575—41no—
81C2×C83⋊C8C2311yescharacteristic
82C2×C83⋊C8C2311yescharacteristic
83C4⋊Dic83—21no—
84C4⋊Dic83C2311yescharacteristic
85Dic83⋊C4—21no—
86Dic83⋊C4—21no—
87C22⋊Dic83—21no—
88C22⋊Dic83—21no—
89(C2×D83)⋊C4—21no—
90(C2×D83)⋊C4—21no—
91(C2×D83)⋊C4—21no—
92(C2×D83)⋊C4—21no—
93(C2×D83)⋊C4—21no—
94(C2×D83)⋊C4—21no—
95(C2×D83)⋊C4—21no—
96(C2×D83)⋊C4—21no—
97(C2×D83)⋊C4—21no—
98C4×Dic83D411yescharacteristic
99C4×Dic83D411yescharacteristic
100C4×Dic83C2311yescharacteristic
101C2×Q8×C83C2311yescharacteristic
102C2×Q8×C83C2311yescharacteristic
103C2×D4×C83C2311yescharacteristic
104Q16×C83—41no—
105SD16×C83—41no—
106C2×C8×C83C2311yescharacteristic
107C2×C8×C83C2311yescharacteristic
108C4⋊C4×C83—21no—
109C4⋊C4×C83C2311yescharacteristic
110C22⋊C4×C83—21no—
111C22⋊C4×C83—21no—
112C42×C83—21no—
113C42×C83C2311yescharacteristic
114C23×C83—21no—
115C22×D83—21no—
116C22×D83—21no—
117C22×D83—42no—
118C22×D83—42no—
119C22×D83—42no—
120C22×D83C2×D411yescharacteristic
121Q8×C83—21no—
122Q8×C83—21no—
123Q8×C83—21no—
124Q8×C83—21no—
125D4×C83—21no—
126D4×C83—21no—
127C2×C4×C83—21no—
128C2×C4×C83—21no—
129C2×C4×C83—21no—
130C2×C4×C83C2×D411yescharacteristic
131C2×C4×C83C2×D411yescharacteristic
132C2×C4×C83C2411yescharacteristic
133C22⋊D83—42no—
134C2×Dic83—21no—
135C2×Dic83—21no—
136C2×Dic83C2×D411yescharacteristic
137C2×Dic83C2×D411yescharacteristic
138C2×Dic83C2×D411yescharacteristic
139D332—21no—
140D332—21no—
141D332—21no—
142D332—21no—
143D332—42no—
144C4×D83—22no—
145C4×D83—21no—
146C4×D83—21no—
147C4×D83—41no—
148C4×D83—42no—
149C4×D83—42no—
150Dic166—21no—
151Dic166—21no—
152Dic166—42no—
153C8×C83—21no—
154C8×C83—21no—
155C83⋊C8—21no—
156C83⋊C8—21no—
157C22×C83—41no—
158C22×C83—41no—
159C22×C83—41no—
160C22×C83C22×D411yescharacteristic
161C2×D83—21no—
162C2×D83—21no—
163C2×D83—41no—
164C2×D83—41no—
165C2×D83—41no—
166C2×D83—41no—
167C2×D83—41no—
168C2×D83—41no—
169C2×D83—84no—
170C2×D83#13512yes—
171C4×C83—21no—
172C4×C83—21no—
173C4×C83—41no—
174C4×C83—41no—
175C4×C83—41no—
176C4×C83H(5,𝔽2)11yescharacteristic
177C4×C83H(5,𝔽2)11yescharacteristic
178Dic83—21no—
179Dic83—21no—
180Dic83—21no—
181Dic83—41no—
182Dic83—41no—
183C2×C83—81no—
184C2×C83#53511yescharacteristic
185C2×C83C2×#13511yescharacteristic
186C2×C83#57511yescharacteristic
187D83—22no—
188D83—81no—
189D83—81no—
190#3211—831nomaximal, Sylow
191C83#321111yescharacteristic, complemented, Sylow
192#529—831no—
193C2×#124—831no—
194#495—831no—
195#495—831no—
196#488—831no—
197#488—831no—
198#464—831no—
199#464D8311yescharacteristic, complemented
200#463—831no—
201#463—831no—
202#451—831no—
203#450—831no—
204#448—831no—
205#447—831no—
206#431—831no—
207C22×Q8—831no—
208C22×D4—831no—
209C2×Q16—831no—
210C2×Q16C2×D8311yescharacteristic, complemented
211C2×SD16—831no—
212C2×SD16—831no—
213C2×SD16—831no—
214C2×SD16C2×D8311yescharacteristic
215C2×D8—831no—
216C2×D8—831no—
217#126—1661no—
218#125—831no—
219#124—831no—
220#124—831no—
221#124—831no—
222#124—831no—
223#124—1661no—
224#124—1661no—
225#124C2×D8311yescharacteristic
226#123—1661no—
227C22⋊Q8—831no—
228#121—831no—
229C4×Q8C2×D8311yescharacteristic, complemented
230C4×D4—831no—
231C4×D4—1661no—
232#117—1661no—
233C2×C22⋊C4—1661no—
234C2×C22⋊C4—1661no—
235C2×C42—831no—
236C4⋊C8—831no—
237C4⋊C8C2×D8311yescharacteristic, complemented
238#103—831no—
239#103—831no—
240#103—831no—
241#103C2×D8311yescharacteristic, complemented
242#102—831no—
243#102—831no—
244#102—831no—
245#102C2×D8311yescharacteristic
246C22⋊C8—831no—
247C22⋊C8—831no—
248C4×C8—831no—
249C4×C8—831no—
250C24—1661no—
251C2×Q8—3321no—
252C2×Q8—3321no—
253C2×Q8—831no—
254C2×Q8—1661no—
255C2×Q8C22×D8311yescharacteristic
256C2×Q8C22×D8311yescharacteristic
257C2×D4—3321no—
258C2×D4—3321no—
259C2×D4—831no—
260C2×D4—831no—
261C2×D4—1661no—
262C2×D4—1661no—
263C2×D4C22×D8311yescharacteristic
264C22×C4—831no—
265C22×C4—831no—
266C22×C4—831no—
267C22×C4—1661no—
268C22×C4—1661no—
269Q16—41no—
270Q16—3321no—
271SD16—41no—
272SD16—3321no—
273SD16—3321no—
274SD16—3321no—
275D8—3321no—
276D8—3321no—
277C2×C8—831no—
278C2×C8—831no—
279C2×C8C22×D8311yescharacteristic
280C2×C8C22×D8311yescharacteristic
281C4⋊C4—21no—
282C4⋊C4—831no—
283C4⋊C4—1661no—
284C4⋊C4—1661no—
285C4⋊C4—1661no—
286C4⋊C4C22×D8311yescharacteristic
287C22⋊C4—21no—
288C22⋊C4—21no—
289C22⋊C4—1661no—
290C22⋊C4—1661no—
291C22⋊C4—1661no—
292C22⋊C4—1661no—
293C22⋊C4—1661no—
294C22⋊C4—1661no—
295C22⋊C4—1661no—
296C22⋊C4—1661no—
297C22⋊C4—1661no—
298C22⋊C4—1661no—
299C22⋊C4—1661no—
300C42—21no—
301C42—831no—
302C42—831no—
303C42—831no—
304C42C22×D8311yescharacteristic
305C23—21no—
306C23—3322no—
307C23—3322no—
308C23—3322no—
309C23—831no—
310C23—1661no—
311C23—1661no—
312Q8—21no—
313Q8—21no—
314Q8—21no—
315Q8—21no—
316Q8—3322no—
317Q8—1661no—
318Q8—1661no—
319D4—21no—
320D4—21no—
321D4—3322no—
322D4—3322no—
323D4—1661no—
324D4—1661no—
325D4—1661no—
326D4—1661no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—3322no—
331C2×C4—3321no—
332C2×C4—3322no—
333C2×C4—831no—
334C2×C4—831no—
335C2×C4—831no—
336C2×C4—1662no—
337C2×C4—1661no—
338C2×C4—1661no—
339C2×C4—1661no—
340C2×C4—1661no—
341C2×C4D4×D8311yescharacteristic
342C2×C4D4×D8311yescharacteristic
343C2×C4C23×D8311yescharacteristic
344C8—21no—
345C8—21no—
346C8—1661no—
347C8—1661no—
348C22—41no—
349C22—41no—
350C22—41no—
351C22—3321no—
352C22—3321no—
353C22—3321no—
354C22—3321no—
355C22—3321no—
356C22—3321no—
357C22—832no—
358C22—6644no—
359C22—1661no—
360C22—1661no—
361C22C2×D4×D8311yescharacteristic
362C4—21no—
363C4—21no—
364C4—41no—
365C4—41no—
366C4—41no—
367C4—3321no—
368C4—3321no—
369C4—1661no—
370C4—1661no—
371C4—1661no—
372C4#1458183711yescharacteristic
373C4#1458183811yescharacteristic
374C2—81no—
375C2—6641no—
376C2—6641no—
377C2—1662no—
378C2#1912826711yescharacteristic
379C2#135×D8311yescharacteristic
380C2#1912850011yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#464×C83C2
socleC22×C83C22×D4
prime radical#19127836C2
hypercenter#464D83
hypocenterC83#3211
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C42 ◁ #464

Complements

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

NH ≅ G/Ntype
1. #19128238C2semidirect
2. #124×D83C2semidirect
3. #19128086C2semidirect
4. #19128082C2semidirect
5. #19128057C2semidirect
6. #19128054C2semidirect
7. #19127923C2semidirect
8. #19127917C2semidirect
9. #19127913C2semidirect
10. #19127855C2semidirect
11. #19127850C2semidirect
12. #19127841C2semidirect
14. #19127753C2semidirect
15. #464×C83C2semidirect
18. C2×#13050580C22semidirect
19. C2×#13050579C22semidirect
21. C2×#13050577C22semidirect
25. #14581783C22semidirect
27. #14581781C22semidirect
36. C4×D332C22semidirect
41. C4⋊(C83⋊C8)C22semidirect
42. #14581717C22semidirect
43. #14581716C22semidirect
44. #14581715C22semidirect
50. C8×Dic83C22semidirect
51. C8×Dic83C22semidirect
52. C2×Q16×C83C22semidirect
55. C4×Q8×C83C22semidirect
56. C4⋊C8×C83C22semidirect
57. #103×C83C22semidirect
191. C83#3211semidirect
199. #464D83semidirect
210. C2×Q16C2×D83semidirect
229. C4×Q8C2×D83semidirect
237. C4⋊C8C2×D83semidirect
241. #103C2×D83semidirect

Maximal subgroups

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

subgroupindex
#191282382
#124×D832
#191280862
#191280822
#191280572
#191280542
#191279232
#191279172
#191279132
#191278552
#191278502
#191278412
#191278362
#191277532
#464×C832
#321183