Finite Group Databasebeta

← #24918331

Subgroups — #24918331

Order 10624.

410 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#19128274C211yesmaximal, characteristic, complemented
2#19128267C211yesmaximal, characteristic
3#19128044C211yesmaximal, characteristic, complemented
4#19128041C211yesmaximal, characteristic, complemented
5#19128028C211yesmaximal, characteristic, complemented
6#19128023C211yesmaximal, characteristic, complemented
7#19127947C211yesmaximal, characteristic, complemented
8#19127932C211yesmaximal, characteristic, complemented
9#19127928C211yesmaximal, characteristic, complemented
10#19127853C211yesmaximal, characteristic, complemented
11#19127850C211yesmaximal, characteristic, complemented
12#19127841C211yesmaximal, characteristic, complemented
13#19127838C211yesmaximal, characteristic, complemented
14#19127669C211yesmaximal, characteristic, complemented
15#468×C83C211yesmaximal, characteristic, complemented
16C2×#13050590C2211yescharacteristic
17C2×Q8×D83C2211yescharacteristic, complemented
18C2×D4×D83C2211yescharacteristic
19C2×#13050579C2211yescharacteristic
20C2×#13050578C2211yescharacteristic
21C2×#13050577C2211yescharacteristic
22C22×C83⋊C8C2211yescharacteristic, complemented
23#14581791—21no—
24#14581788—21no—
25#14581778—21no—
26#14581778—21no—
27#14581777—21no—
28#14581774—21no—
29#14581772C2211yescharacteristic
30#14581771C2211yescharacteristic, complemented
31#14581768C2211yescharacteristic
32#14581767C2211yescharacteristic
33#14581766C2211yescharacteristic
34#14581763C2211yescharacteristic
35#14581762—21no—
36#14581760—21no—
37#14581750C2211yescharacteristic
38#14581750C2211yescharacteristic, complemented
39#14581726C2211yescharacteristic, complemented
40#14581723C2211yescharacteristic, complemented
41#14581723C2211yescharacteristic, complemented
42#14581717C2211yescharacteristic, complemented
43#14581716C2211yescharacteristic, complemented
44#14581715C2211yescharacteristic, complemented
45#14581714C2211yescharacteristic
46#14581713C2211yescharacteristic
47#14581712C2211yescharacteristic
48(C2×D83)⋊C8C2211yescharacteristic
49(C2×D83)⋊C8C2211yescharacteristic
50(C2×D83)⋊C8C2211yescharacteristic, complemented
51(C2×D83)⋊C8C2211yescharacteristic, complemented
52C2×SD16×C83C2211yescharacteristic
53C2×M16×C83C2211yescharacteristic, complemented
54C22⋊Q8×C83C2211yescharacteristic, complemented
55#121×C83C2211yescharacteristic
56#107×C83C2211yescharacteristic, complemented
57#103×C83C2211yescharacteristic, complemented
58#102×C83C2211yescharacteristic
59C23×D83—21no—
60#13050590—41no—
61#13050590—41no—
62C2×Dic166—21no—
63C2×Dic166D411yescharacteristic
64C2×Dic166C2311yescharacteristic
65Q8×D83—41no—
66Q8×D83—41no—
67C2×D332—21no—
68C2×D332D411yescharacteristic
69C2×D332C2311yescharacteristic
70C2×C22⋊D83—21no—
71C2×C22⋊D83—21no—
72C2×C22⋊D83—21no—
73C2×C22⋊D83D411yescharacteristic
74C2×C22⋊D83D411yescharacteristic
75D4×D83—41no—
76D4×D83—41no—
77C2×C4×D83—21no—
78C2×C4×D83C2311yescharacteristic
79C2×C4×D83C2311yescharacteristic
80#13050579—41no—
81#13050578—41no—
82#13050577—41no—
83C2×C83⋊C8—41no—
84C2×C83⋊C8C2311yescharacteristic
85C2×C83⋊C8C2311yescharacteristic
86C4⋊Dic83—21no—
87C4⋊Dic83C2311yescharacteristic
88C4⋊Dic83C2311yescharacteristic
89Dic83⋊C4—21no—
90Dic83⋊C4—21no—
91C22⋊Dic83—21no—
92C22⋊Dic83—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—
98(C2×D83)⋊C4—21no—
99C4×Dic83—21no—
100C4×Dic83—21no—
101C2×Q8×C83C2311yescharacteristic
102C2×D4×C83—21no—
103C2×D4×C83C2311yescharacteristic
104C22×C4×C83C2311yescharacteristic
105SD16×C83—41no—
106M16×C83—41no—
107C2×C8×C83C2311yescharacteristic
108C2×C8×C83C2311yescharacteristic
109C4⋊C4×C83—21no—
110C4⋊C4×C83C2311yescharacteristic
111C4⋊C4×C83C2311yescharacteristic
112C22⋊C4×C83—21no—
113C22⋊C4×C83—21no—
114C23×C83—21no—
115C23×C83C2×D411yescharacteristic
116C22×D83—21no—
117C22×D83—42no—
118C22×D83—81no—
119C22×D83—81no—
120C22×D83C2×D411yescharacteristic
121C22×D83C2×D411yescharacteristic
122Q8×C83—21no—
123Q8×C83—21no—
124D4×C83—21no—
125D4×C83—21no—
126D4×C83—81no—
127C2×C4×C83—21no—
128C2×C4×C83—21no—
129C2×C4×C83—21no—
130C2×C4×C83—21no—
131C2×C4×C83—21no—
132C2×C4×C83C2×D411yescharacteristic
133C2×C4×C83C2411yescharacteristic
134C22⋊D83—41no—
135C22⋊D83—41no—
136C22⋊D83—42no—
137C22⋊D83—81no—
138C22⋊D83—81no—
139C2×Dic83—21no—
140C2×Dic83—21no—
141C2×Dic83—21no—
142C2×Dic83C2×D411yescharacteristic
143C2×Dic83C2×D411yescharacteristic
144D332—21no—
145D332—21no—
146D332—41no—
147D332—81no—
148C4×D83—21no—
149C4×D83—21no—
150C4×D83—21no—
151C4×D83—21no—
152C4×D83—42no—
153Dic166—21no—
154Dic166—21no—
155Dic166—41no—
156Dic166—42no—
157C8×C83—21no—
158C8×C83—21no—
159C83⋊C8—21no—
160C83⋊C8—21no—
161C22×C83—21no—
162C22×C83—21no—
163C22×C83—21no—
164C22×C83—41no—
165C22×C83—41no—
166C22×C83—41no—
167C22×C83C22×D411yescharacteristic
168C2×D83—21no—
169C2×D83—21no—
170C2×D83—21no—
171C2×D83—21no—
172C2×D83—21no—
173C2×D83—21no—
174C2×D83—41no—
175C2×D83—41no—
176C2×D83—41no—
177C2×D83—84no—
178C4×C83—21no—
179C4×C83—41no—
180C4×C83—41no—
181C4×C83—41no—
182C4×C83H(5,𝔽2)11yescharacteristic
183C4×C83H(5,𝔽2)11yescharacteristic
184Dic83—21no—
185Dic83—21no—
186Dic83—41no—
187Dic83—41no—
188Dic83—41no—
189C2×C83—41no—
190C2×C83—81no—
191C2×C83#53411yescharacteristic
192C2×C83#57711yescharacteristic
193C2×C83#57711yescharacteristic
194D83—41no—
195D83—41no—
196D83—81no—
197#3200—831nomaximal, Sylow
198C83#320011yescharacteristic, complemented, Sylow
199#536—831no—
200#535—831no—
201#485—831no—
202#485—831no—
203#482—831no—
204#482—831no—
205#468—831no—
206#468D8311yescharacteristic, complemented
207#465—831no—
208#465—831no—
209#450—831no—
210#450—831no—
211#448—831no—
212#448—831no—
213#408—831no—
214C2×#41—831no—
215C22×Q8—831no—
216C22×D4—831no—
217C2×SD16—831no—
218C2×SD16—831no—
219C2×SD16—831no—
220C2×SD16C2×D8311yescharacteristic
221C2×M16C2×D8311yescharacteristic, complemented
222C22×C8—831no—
223C4⋊Q8—1661no—
224#127—1661no—
225#124—1661no—
226#124—1661no—
227#123—1661no—
228#123—1661no—
229C22⋊Q8—831no—
230C22⋊Q8—831no—
231C22⋊Q8—831no—
232C22⋊Q8C2×D8311yescharacteristic, complemented
233#121—831no—
234#121—831no—
235#121—831no—
236#121C2×D8311yescharacteristic
237#120—1661no—
238#120—1661no—
239#117—831no—
240#117—831no—
241#107—831no—
242#107C2×D8311yescharacteristic, complemented
243#106—831no—
244#106—831no—
245#103—831no—
246#103—831no—
247#103—831no—
248#103C2×D8311yescharacteristic, complemented
249#102—831no—
250#102—831no—
251#102—831no—
252#102C2×D8311yescharacteristic
253C22⋊C8—831no—
254C22⋊C8—831no—
255C22⋊C8—831no—
256C22⋊C8—831no—
257C24—1661no—
258#41—3321no—
259#41—3321no—
260C2×Q8—3321no—
261C2×Q8—3321no—
262C2×Q8—831no—
263C2×Q8—831no—
264C2×Q8—1661no—
265C2×Q8C22×D8311yescharacteristic
266C2×D4—21no—
267C2×D4—3321no—
268C2×D4—3321no—
269C2×D4—831no—
270C2×D4—831no—
271C2×D4—831no—
272C2×D4—831no—
273C2×D4—1661no—
274C2×D4—1661no—
275C2×D4—1661no—
276C2×D4—1661no—
277C2×D4C22×D8311yescharacteristic
278C22×C4—831no—
279C22×C4—831no—
280C22×C4—1661no—
281C22×C4C22×D8311yescharacteristic
282SD16—41no—
283SD16—3321no—
284SD16—3321no—
285SD16—3321no—
286M16—41no—
287C2×C8—3321no—
288C2×C8—831no—
289C2×C8—831no—
290C2×C8C22×D8311yescharacteristic
291C2×C8C22×D8311yescharacteristic
292C4⋊C4—21no—
293C4⋊C4—831no—
294C4⋊C4—831no—
295C4⋊C4—1661no—
296C4⋊C4—1661no—
297C4⋊C4—1661no—
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—1661no—
311C42—1661no—
312C23—21no—
313C23—3322no—
314C23—831no—
315C23—831no—
316C23—6641no—
317C23—6641no—
318C23—1661no—
319C23D4×D8311yescharacteristic
320Q8—21no—
321Q8—21no—
322Q8—3321no—
323Q8—3322no—
324Q8—1661no—
325Q8—1661no—
326D4—21no—
327D4—21no—
328D4—81no—
329D4—3321no—
330D4—3321no—
331D4—3321no—
332D4—3322no—
333D4—6641no—
334D4—6641no—
335D4—6641no—
336D4—1661no—
337D4—1661no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—3322no—
344C2×C4—831no—
345C2×C4—831no—
346C2×C4—1661no—
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—41no—
363C22—41no—
364C22—41no—
365C22—3321no—
366C22—3321no—
367C22—3321no—
368C22—6644no—
369C22—1661no—
370C22—1661no—
371C22—1661no—
372C22—1661no—
373C22—1661no—
374C22—1661no—
375C22C2×D4×D8311yescharacteristic
376C4—21no—
377C4—41no—
378C4—41no—
379C4—41no—
380C4—3321no—
381C4—3321no—
382C4—3321no—
383C4—1661no—
384C4—1661no—
385C4#1458183711yescharacteristic
386C4#1458183811yescharacteristic
387C2—41no—
388C2—81no—
389C2—3321no—
390C2—3321no—
391C2—6641no—
392C2#1912825911yescharacteristic
393C2#1912851011yescharacteristic
394C2#1912851111yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#468×C83C2
socleC22×C83C22×D4
prime radical#19128267C2
hypercenter#468D83
hypocenterC83#3200
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ #468

Complements

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

NH ≅ G/Ntype
1. #19128274C2semidirect
3. #19128044C2semidirect
4. #19128041C2semidirect
5. #19128028C2semidirect
6. #19128023C2semidirect
7. #19127947C2semidirect
8. #19127932C2semidirect
9. #19127928C2semidirect
10. #19127853C2semidirect
11. #19127850C2semidirect
12. #19127841C2semidirect
13. #19127838C2semidirect
14. #19127669C2semidirect
15. #468×C83C2semidirect
17. C2×Q8×D83C22semidirect
22. C22×C83⋊C8C22semidirect
30. #14581771C22semidirect
38. #14581750C22semidirect
39. #14581726C22semidirect
40. #14581723C22semidirect
41. #14581723C22semidirect
42. #14581717C22semidirect
43. #14581716C22semidirect
44. #14581715C22semidirect
50. (C2×D83)⋊C8C22semidirect
51. (C2×D83)⋊C8C22semidirect
53. C2×M16×C83C22semidirect
54. C22⋊Q8×C83C22semidirect
56. #107×C83C22semidirect
57. #103×C83C22semidirect
198. C83#3200semidirect
206. #468D83semidirect
221. C2×M16C2×D83semidirect
232. C22⋊Q8C2×D83semidirect
242. #107C2×D83semidirect
248. #103C2×D83semidirect

Maximal subgroups

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

subgroupindex
#191282742
#191282672
#191280442
#191280412
#191280282
#191280232
#191279472
#191279322
#191279282
#191278532
#191278502
#191278412
#191278382
#191276692
#468×C832
#320083