Finite Group Databasebeta

← #24919701

Subgroups — #24919701

Order 10624.

460 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#19128414C211yesmaximal, characteristic, complemented
2D4×C22⋊D83C211yesmaximal, characteristic, complemented
3#19128046C211yesmaximal, characteristic, complemented
4#19128025C211yesmaximal, characteristic, complemented
5#19128024C211yesmaximal, characteristic, complemented
6#19127989C211yesmaximal, characteristic, complemented
7#19127954C211yesmaximal, characteristic, complemented
8#19127953C211yesmaximal, characteristic, complemented
9#19127895C211yesmaximal, characteristic, complemented
10#19127836C211yesmaximal, characteristic, complemented
11#19127835C211yesmaximal, characteristic, complemented
12#19127781C211yesmaximal, characteristic, complemented
13#19127777C211yesmaximal, characteristic, complemented
14#19127704C211yesmaximal, characteristic, complemented
15#418×C83C211yesmaximal, characteristic, complemented
16C22×C22⋊D83—21no—
17C2×D4×D83C2211yescharacteristic
18C2×D664C2211yescharacteristic, complemented
19C2×#13050575C2211yescharacteristic
20C2×#13050575C2211yescharacteristic, complemented
21C2×#13050573C2211yescharacteristic, complemented
22#14581791C2211yescharacteristic, complemented
23#14581788C2211yescharacteristic
24#14581780—21no—
25#14581778—21no—
26#14581767C2211yescharacteristic, complemented
27#14581766C2211yescharacteristic
28#14581766C2211yescharacteristic, complemented
29#14581765—21no—
30#14581764—21no—
31#14581763C2211yescharacteristic
32#14581761—21no—
33#14581760—21no—
34D4×Dic83C2211yescharacteristic
35#14581754C2211yescharacteristic, complemented
36C4×C22⋊D83C2211yescharacteristic
37#14581750C2211yescharacteristic, complemented
38#14581749—21no—
39#14581725C2211yescharacteristic, complemented
40#14581724C2211yescharacteristic, complemented
41#14581723C2211yescharacteristic, complemented
42Dic83⋊C8C2211yescharacteristic, complemented
43#14581714C2211yescharacteristic, complemented
44#14581714C2211yescharacteristic, complemented
45#14581713C2211yescharacteristic
46#14581713C2211yescharacteristic, complemented
47#14581712C2211yescharacteristic, complemented
48#14581712C2211yescharacteristic, complemented
49C22⋊(C83⋊C8)C2211yescharacteristic, complemented
50(C2×D83)⋊C8C2211yescharacteristic, complemented
51C8×Dic83C2211yescharacteristic, complemented
52C22×D4×C83C2211yescharacteristic
53C2×M16×C83C2211yescharacteristic, complemented
54#117×C83C2211yescharacteristic, complemented
55#102×C83C2211yescharacteristic
56#102×C83C2211yescharacteristic
57#102×C83C2211yescharacteristic, complemented
58#102×C83C2211yescharacteristic, complemented
59C23×D83—21no—
60C2×Dic166—21no—
61C2×D332—21no—
62C2×D332C2311yescharacteristic
63C2×D332C2311yescharacteristic
64C2×C22⋊D83—21no—
65C2×C22⋊D83—21no—
66C2×C22⋊D83—21no—
67C2×C22⋊D83—21no—
68C2×C22⋊D83—21no—
69C2×C22⋊D83—21no—
70C2×C22⋊D83—44no—
71C2×C22⋊D83D411yescharacteristic
72D4×D83—22no—
73D4×D83—41no—
74C22×Dic83—21no—
75C2×C4×D83—21no—
76C2×C4×D83C2311yescharacteristic
77D664—41no—
78#13050575—22no—
79#13050575—41no—
80#13050573—41no—
81C2×C83⋊C8C2311yescharacteristic
82C2×C83⋊C8C2311yescharacteristic
83C4⋊Dic83C2311yescharacteristic
84C4⋊Dic83C2311yescharacteristic
85Dic83⋊C4—21no—
86Dic83⋊C4D411yescharacteristic
87C22⋊Dic83—21no—
88C22⋊Dic83—21no—
89C22⋊Dic83D411yescharacteristic
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)⋊C4D411yescharacteristic
96C4×Dic83—21no—
97C4×Dic83C2311yescharacteristic
98C24×C83—21no—
99C2×D4×C83—22no—
100C2×D4×C83—21no—
101C2×D4×C83—41no—
102C2×D4×C83C2311yescharacteristic
103C2×D4×C83C2311yescharacteristic
104C22×C4×C83C2311yescharacteristic
105M16×C83—41no—
106C2×C8×C83C2311yescharacteristic
107C2×C8×C83C2311yescharacteristic
108C4⋊C4×C83C2311yescharacteristic
109C4⋊C4×C83C2311yescharacteristic
110C22⋊C4×C83—21no—
111C42×C83—21no—
112C23×C83—21no—
113C23×C83—21no—
114C23×C83—42no—
115C23×C83—42no—
116C23×C83—81no—
117C23×C83C2×D411yescharacteristic
118C22×D83—21no—
119C22×D83—21no—
120C22×D83—42no—
121C22×D83—42no—
122C22×D83—81no—
123C22×D83C2×D411yescharacteristic
124D4×C83—21no—
125D4×C83—21no—
126D4×C83—21no—
127D4×C83—42no—
128D4×C83#4112yes—
129C2×C4×C83—21no—
130C2×C4×C83—21no—
131C2×C4×C83—21no—
132C2×C4×C83—21no—
133C2×C4×C83C2×D411yescharacteristic
134C2×C4×C83C2411yescharacteristic
135C22⋊D83—22no—
136C22⋊D83—42no—
137C22⋊D83—42no—
138C22⋊D83—42no—
139C22⋊D83—42no—
140C22⋊D83—81no—
141C22⋊D83—81no—
142C2×Dic83—21no—
143C2×Dic83—21no—
144C2×Dic83—42no—
145C2×Dic83C2×D411yescharacteristic
146C2×Dic83C2×D411yescharacteristic
147C2×Dic83C2×D411yescharacteristic
148D332—21no—
149D332—21no—
150D332—21no—
151D332—21no—
152D332—81no—
153C4×D83—21no—
154C4×D83—21no—
155C4×D83—42no—
156Dic166—81no—
157C8×C83—21no—
158C8×C83—21no—
159C83⋊C8—21no—
160C83⋊C8—21no—
161C22×C83—22no—
162C22×C83—21no—
163C22×C83—21no—
164C22×C83—21no—
165C22×C83—41no—
166C22×C83—41no—
167C22×C83—41no—
168C22×C83—41no—
169C22×C83—41no—
170C22×C83—84no—
171C22×C83C22×D411yescharacteristic
172C2×D83—21no—
173C2×D83—21no—
174C2×D83—21no—
175C2×D83—41no—
176C2×D83—41no—
177C2×D83—41no—
178C2×D83—41no—
179C2×D83—41no—
180C2×D83—41no—
181C2×D83—84no—
182C4×C83—21no—
183C4×C83—41no—
184C4×C83—41no—
185C4×C83C2×#4111yescharacteristic
186C4×C83H(5,𝔽2)11yescharacteristic
187Dic83—21no—
188Dic83—21no—
189Dic83—41no—
190Dic83—41no—
191Dic83AGL(1,ℤ/8)12yes—
192C2×C83—42no—
193C2×C83—41no—
194C2×C83—81no—
195C2×C83#54611yescharacteristic
196C2×C83C2×AGL(1,ℤ/8)11yescharacteristic
197C2×C83#57611yescharacteristic
198D83—41no—
199D83—81no—
200D83—81no—
201#3325—831nomaximal, Sylow
202C83#332511yescharacteristic, complemented, Sylow
203#555—831no—
204D42—831no—
205#486—831no—
206#482—831no—
207#482—831no—
208#476—831no—
209#469—831no—
210#469—831no—
211#459—831no—
212#447—831no—
213#447—831no—
214C4×D8—831no—
215#436—831no—
216#418—831no—
217#418D8311yescharacteristic, complemented
218C22×D4—831no—
219C22×D4—1661no—
220C22×D4C2×D8311yescharacteristic
221C2×D8—831no—
222C2×D8—831no—
223C2×D8—831no—
224C2×M16—831no—
225C2×M16C2×D8311yescharacteristic, complemented
226C4⋊Q8—831no—
227#127—831no—
228#124—1661no—
229#124—1661no—
230#121—831no—
231#121—831no—
232#121—831no—
233#121—831no—
234#121—1661no—
235#121—1661no—
236#120—1661no—
237#120—1661no—
238C4×D4—831no—
239C4×D4—831no—
240C4×D4—831no—
241#117—831no—
242#117—1661no—
243#117C2×D8311yescharacteristic, complemented
244#107—831no—
245#106—831no—
246#106—831no—
247C4⋊C8—831no—
248#102—831no—
249#102—831no—
250#102—831no—
251#102—831no—
252#102—831no—
253#102—831no—
254#102C2×D8311yescharacteristic
255#102C2×D8311yescharacteristic
256#102C2×D8311yescharacteristic, complemented
257#102C2×D8311yescharacteristic, complemented
258C22⋊C8—831no—
259C22⋊C8—831no—
260C4×C8—831no—
261C24—21no—
262C24—1661no—
263C2×Q8—1661no—
264C2×D4—22no—
265C2×D4—21no—
266C2×D4—41no—
267C2×D4—3324no—
268C2×D4—3321no—
269C2×D4—831no—
270C2×D4—831no—
271C2×D4—831no—
272C2×D4—1661no—
273C2×D4—1661no—
274C2×D4—1661no—
275C2×D4—1661no—
276C2×D4—1661no—
277C2×D4—1661no—
278C2×D4—1662no—
279C2×D4—1661no—
280C2×D4C22×D8311yescharacteristic
281C2×D4C22×D8311yescharacteristic
282C22×C4—831no—
283C22×C4—1661no—
284C22×C4—1661no—
285C22×C4C22×D8311yescharacteristic
286D8—3321no—
287D8—3321no—
288D8—1662no—
289M16—41no—
290M16—3321no—
291C2×C8—831no—
292C2×C8—831no—
293C2×C8C22×D8311yescharacteristic
294C2×C8C22×D8311yescharacteristic
295C4⋊C4—831no—
296C4⋊C4—831no—
297C4⋊C4—831no—
298C4⋊C4—1661no—
299C4⋊C4C22×D8311yescharacteristic
300C4⋊C4C22×D8311yescharacteristic
301C22⋊C4—21no—
302C22⋊C4—831no—
303C22⋊C4—831no—
304C22⋊C4—1661no—
305C22⋊C4—1661no—
306C22⋊C4—1661no—
307C22⋊C4—1661no—
308C22⋊C4—1661no—
309C22⋊C4—1661no—
310C22⋊C4—1661no—
311C42—21no—
312C42—831no—
313C42—1661no—
314C23—21no—
315C23—21no—
316C23—42no—
317C23—42no—
318C23—81no—
319C23—3322no—
320C23—3322no—
321C23—831no—
322C23—6641no—
323C23—1661no—
324C23—1661no—
325C23D4×D8311yescharacteristic
326Q8—6641no—
327D4—21no—
328D4—21no—
329D4—21no—
330D4—42no—
331D4—3322no—
332D4—3322no—
333D4—3322no—
334D4—3322no—
335D4—6641no—
336D4—6641no—
337D4—6641no—
338D4—1662no—
339D4—1661no—
340D4—1661no—
341D4—1661no—
342D4—1661no—
343D4#1305059012yes—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—3322no—
349C2×C4—3322no—
350C2×C4—831no—
351C2×C4—831no—
352C2×C4—831no—
353C2×C4—1661no—
354C2×C4—1661no—
355C2×C4—1661no—
356C2×C4—1661no—
357C2×C4D4×D8311yescharacteristic
358C2×C4C23×D8311yescharacteristic
359C8—21no—
360C8—21no—
361C8—1661no—
362C8—1661no—
363C22—21no—
364C22—22no—
365C22—21no—
366C22—21no—
367C22—41no—
368C22—41no—
369C22—41no—
370C22—41no—
371C22—41no—
372C22—84no—
373C22—3321no—
374C22—3321no—
375C22—3321no—
376C22—3321no—
377C22—3321no—
378C22—3321no—
379C22—6644no—
380C22—1661no—
381C22—1661no—
382C22—1661no—
383C22C2×D4×D8311yescharacteristic
384C4—21no—
385C4—41no—
386C4—41no—
387C4—3321no—
388C4—3321no—
389C4—832no—
390C4—1661no—
391C4—1661no—
392C4C2×#1305059011yescharacteristic
393C4#1458183711yescharacteristic
394C2—41no—
395C2—42no—
396C2—81no—
397C2—3321no—
398C2—6641no—
399C2—6641no—
400C2#1912834111yescharacteristic
401C2AGL(1,ℤ/8)×D8311yescharacteristic
402C2#1912850511yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#418×C83C2
socleC22×C83C22×D4
hypercenter#418D83
hypocenterC83#3325
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ #418

Complements

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

NH ≅ G/Ntype
1. #19128414C2semidirect
2. D4×C22⋊D83C2semidirect
3. #19128046C2semidirect
4. #19128025C2semidirect
5. #19128024C2semidirect
6. #19127989C2semidirect
7. #19127954C2semidirect
8. #19127953C2semidirect
9. #19127895C2semidirect
10. #19127836C2semidirect
11. #19127835C2semidirect
12. #19127781C2semidirect
13. #19127777C2semidirect
14. #19127704C2semidirect
15. #418×C83C2semidirect
18. C2×D664C22semidirect
20. C2×#13050575C22semidirect
21. C2×#13050573C22semidirect
22. #14581791C22semidirect
26. #14581767C22semidirect
28. #14581766C22semidirect
35. #14581754C22semidirect
37. #14581750C22semidirect
39. #14581725C22semidirect
40. #14581724C22semidirect
41. #14581723C22semidirect
42. Dic83⋊C8C22semidirect
43. #14581714C22semidirect
44. #14581714C22semidirect
46. #14581713C22semidirect
47. #14581712C22semidirect
48. #14581712C22semidirect
49. C22⋊(C83⋊C8)C22semidirect
50. (C2×D83)⋊C8C22semidirect
51. C8×Dic83C22semidirect
53. C2×M16×C83C22semidirect
54. #117×C83C22semidirect
57. #102×C83C22semidirect
58. #102×C83C22semidirect
202. C83#3325semidirect
217. #418D83semidirect
225. C2×M16C2×D83semidirect
243. #117C2×D83semidirect
256. #102C2×D83semidirect
257. #102C2×D83semidirect

Maximal subgroups

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

subgroupindex
#191284142
D4×C22⋊D832
#191280462
#191280252
#191280242
#191279892
#191279542
#191279532
#191278952
#191278362
#191278352
#191277812
#191277772
#191277042
#418×C832
#332583