Finite Group Databasebeta

← #24919306

Subgroups — #24919306

Order 10624.

420 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#19128398C211yesmaximal, characteristic, complemented
2#19128339C211yesmaximal, characteristic, complemented
3#19128057C211yesmaximal, characteristic, complemented
4#19128035C211yesmaximal, characteristic, complemented
5#19128026C211yesmaximal, characteristic, complemented
6#19128003C211yesmaximal, characteristic, complemented
7#19127935C211yesmaximal, characteristic, complemented
8#19127927C211yesmaximal, characteristic, complemented
9#19127918C211yesmaximal, characteristic, complemented
10#19127833C211yesmaximal, characteristic, complemented
11D8×Dic83C211yesmaximal, characteristic, complemented
12(C22⋊D83)⋊C8C211yesmaximal, characteristic, complemented
13#19127697C211yesmaximal, characteristic, complemented
14#102×D83C211yesmaximal, characteristic, complemented
15#449×C83C211yesmaximal, characteristic, complemented
16C2×D4×D83C2211yescharacteristic
17C2×#13050575C2211yescharacteristic
18C22×C83⋊C8C2211yescharacteristic, complemented
19C2×C8×D83C2211yescharacteristic
20#14581785—21no—
21#14581783C2211yescharacteristic, complemented
22#14581781C2211yescharacteristic
23#14581775—21no—
24#14581773—21no—
25#14581768C2211yescharacteristic
26#14581766C2211yescharacteristic
27#14581764—21no—
28#14581763—21no—
29#14581763C2211yescharacteristic
30#14581760—21no—
31D4×Dic83C2211yescharacteristic
32D4×Dic83C2211yescharacteristic, complemented
33C4×C22⋊D83C2211yescharacteristic
34(C22⋊D83)⋊C4—21no—
35#14581751C2211yescharacteristic, complemented
36C4⋊C4×D83C2211yescharacteristic
37C2×(C2×D83)⋊C4—21no—
38#14581726C2211yescharacteristic, complemented
39#14581725C2211yescharacteristic, complemented
40#14581724C2211yescharacteristic, complemented
41Dic83⋊C8C2211yescharacteristic, complemented
42#14581716C2211yescharacteristic, complemented
43#14581716C2211yescharacteristic, complemented
44#14581714C2211yescharacteristic
45#14581714C2211yescharacteristic
46#14581713C2211yescharacteristic
47#14581713C2211yescharacteristic
48#14581713C2211yescharacteristic, complemented
49#14581712C2211yescharacteristic
50(C2×D83)⋊C8C2211yescharacteristic
51C8×Dic83C2211yescharacteristic, complemented
52C2×#41×C83C2211yescharacteristic
53C2×SD16×C83C2211yescharacteristic, complemented
54C2×D8×C83C2211yescharacteristic
55#121×C83C2211yescharacteristic, complemented
56#103×C83C2211yescharacteristic, complemented
57#102×C83C2211yescharacteristic
58C22⋊C8×C83C2211yescharacteristic, complemented
59C23×D83—21no—
60C2×D332C2311yescharacteristic
61C2×C22⋊D83—21no—
62C2×C22⋊D83—21no—
63C2×C22⋊D83—21no—
64C2×C22⋊D83D411yescharacteristic
65D4×D83—22no—
66D4×D83—41no—
67C22×Dic83—21no—
68C22×Dic83—21no—
69C2×C4×D83—21no—
70C2×C4×D83C2311yescharacteristic
71#13050575—41no—
72C2×C83⋊C8—41no—
73C2×C83⋊C8C2311yescharacteristic
74C2×C83⋊C8C2311yescharacteristic
75C8×D83—41no—
76C4⋊Dic83—21no—
77C4⋊Dic83C2311yescharacteristic
78C4⋊Dic83C2311yescharacteristic
79C4⋊Dic83C2311yescharacteristic
80Dic83⋊C4—21no—
81Dic83⋊C4—21no—
82Dic83⋊C4—21no—
83Dic83⋊C4D411yescharacteristic
84C22⋊Dic83—21no—
85C22⋊Dic83—21no—
86C22⋊Dic83—21no—
87C22⋊Dic83—21no—
88C22⋊Dic83D411yescharacteristic
89(C2×D83)⋊C4—21no—
90(C2×D83)⋊C4—21no—
91(C2×D83)⋊C4—21no—
92(C2×D83)⋊C4—21no—
93(C2×D83)⋊C4D411yescharacteristic
94C4×Dic83—21no—
95C4×Dic83C2311yescharacteristic
96#41×C83—22no—
97#41×C83—41no—
98C2×Q8×C83C2311yescharacteristic
99C2×D4×C83—21no—
100C2×D4×C83—21no—
101C2×D4×C83C2311yescharacteristic
102C2×D4×C83C2311yescharacteristic
103C22×C4×C83—21no—
104C22×C4×C83C2311yescharacteristic
105SD16×C83—41no—
106D8×C83—22no—
107C2×C8×C83C2311yescharacteristic
108C2×C8×C83C2311yescharacteristic
109C4⋊C4×C83C2311yescharacteristic
110C22⋊C4×C83—21no—
111C23×C83—21no—
112C23×C83—21no—
113C23×C83C2×D411yescharacteristic
114C22×D83—21no—
115C22×D83—42no—
116C22×D83—42no—
117C22×D83—81no—
118C22×D83C2×D411yescharacteristic
119Q8×C83—21no—
120Q8×C83—21no—
121D4×C83—21no—
122D4×C83—21no—
123D4×C83—21no—
124D4×C83—42no—
125D4×C83—81no—
126D4×C83#4112yes—
127C2×C4×C83—21no—
128C2×C4×C83—21no—
129C2×C4×C83—21no—
130C2×C4×C83—21no—
131C2×C4×C83—42no—
132C2×C4×C83C2×D411yescharacteristic
133C2×C4×C83C2411yescharacteristic
134C22⋊D83—22no—
135C22⋊D83—42no—
136C22⋊D83—42no—
137C22⋊D83—81no—
138C2×Dic83—21no—
139C2×Dic83—21no—
140C2×Dic83—21no—
141C2×Dic83—81no—
142C2×Dic83—81no—
143C2×Dic83C2×D411yescharacteristic
144C2×Dic83C2×D411yescharacteristic
145C2×Dic83C2×D411yescharacteristic
146D332—21no—
147D332—21no—
148C4×D83—21no—
149C4×D83—21no—
150C4×D83—81no—
151C8×C83—21no—
152C8×C83—21no—
153C83⋊C8—21no—
154C83⋊C8—21no—
155C22×C83—22no—
156C22×C83—21no—
157C22×C83—21no—
158C22×C83—21no—
159C22×C83—41no—
160C22×C83—41no—
161C22×C83—41no—
162C22×C83—41no—
163C22×C83—41no—
164C22×C83C22×D411yescharacteristic
165C2×D83—21no—
166C2×D83—21no—
167C2×D83—41no—
168C2×D83—41no—
169C2×D83—41no—
170C2×D83—84no—
171C2×D83#13512yes—
172C4×C83—21no—
173C4×C83—41no—
174C4×C83—41no—
175C4×C83C2×#4111yescharacteristic
176C4×C83H(5,𝔽2)11yescharacteristic
177Dic83—21no—
178Dic83—21no—
179Dic83—21no—
180Dic83—41no—
181Dic83—41no—
182Dic83—41no—
183C2×C83—42no—
184C2×C83—41no—
185C2×C83—81no—
186C2×C83#54611yescharacteristic
187C2×C83C2×#13511yescharacteristic
188C2×C83#57611yescharacteristic
189D83—22no—
190D83—81no—
191#3297—831nomaximal, Sylow
192C83#329711yescharacteristic, complemented, Sylow
193#553—831no—
194#546—831no—
195#488—831no—
196#484—831no—
197#482—831no—
198#478—831no—
199#466—831no—
200#465—831no—
201#463—831no—
202#449D8311yescharacteristic, complemented
203#447—831no—
204C4×D8—831no—
205C8×D4—831no—
206#416—831no—
207C2×#102—831no—
208C2×#41C2×D8311yescharacteristic
209C22×D4—831no—
210C2×SD16C2×D8311yescharacteristic, complemented
211C2×D8—831no—
212C2×D8C2×D8311yescharacteristic
213C22×C8—831no—
214C22×C8—831no—
215#126—1661no—
216#125—831no—
217#124—831no—
218#123—1661no—
219#123—1661no—
220C22⋊Q8—831no—
221#121—831no—
222#121—831no—
223#121—1661no—
224#121—1661no—
225#121C2×D8311yescharacteristic, complemented
226#120—1661no—
227C4×D4—831no—
228C4×D4—831no—
229C4×D4—831no—
230C4×D4—1661no—
231#117—831no—
232C2×C4⋊C4—831no—
233C2×C22⋊C4—1661no—
234#107—831no—
235#107—831no—
236#106—831no—
237C4⋊C8—831no—
238#103—831no—
239#103—831no—
240#103C2×D8311yescharacteristic, complemented
241#102—831no—
242#102—831no—
243#102—831no—
244#102—831no—
245#102—831no—
246#102—831no—
247#102C2×D8311yescharacteristic
248C22⋊C8—831no—
249C22⋊C8C2×D8311yescharacteristic, complemented
250C4×C8—831no—
251C24—1661no—
252#41—22no—
253#41—41no—
254C2×Q8C22×D8311yescharacteristic
255C2×D4—21no—
256C2×D4—21no—
257C2×D4—3321no—
258C2×D4—831no—
259C2×D4—831no—
260C2×D4—1661no—
261C2×D4—1661no—
262C2×D4—1662no—
263C2×D4—1661no—
264C2×D4C22×D8311yescharacteristic
265C2×D4C22×D8311yescharacteristic
266C22×C4—21no—
267C22×C4—831no—
268C22×C4—1661no—
269C22×C4—1661no—
270C22×C4—1661no—
271C22×C4C22×D8311yescharacteristic
272SD16—41no—
273D8—22no—
274D8—3321no—
275C2×C8—3321no—
276C2×C8—3321no—
277C2×C8—831no—
278C2×C8—831no—
279C2×C8C22×D8311yescharacteristic
280C2×C8C22×D8311yescharacteristic
281C4⋊C4—831no—
282C4⋊C4—831no—
283C4⋊C4—831no—
284C4⋊C4—831no—
285C4⋊C4—1661no—
286C4⋊C4—1661no—
287C4⋊C4—1661no—
288C4⋊C4—1661no—
289C4⋊C4C22×D8311yescharacteristic
290C22⋊C4—21no—
291C22⋊C4—831no—
292C22⋊C4—831no—
293C22⋊C4—1661no—
294C22⋊C4—1661no—
295C22⋊C4—1661no—
296C22⋊C4—1661no—
297C22⋊C4—1661no—
298C22⋊C4—1661no—
299C22⋊C4—1661no—
300C22⋊C4—1661no—
301C42—831no—
302C42—1661no—
303C23—21no—
304C23—21no—
305C23—3322no—
306C23—3322no—
307C23—831no—
308C23—6641no—
309C23—1661no—
310C23D4×D8311yescharacteristic
311Q8—21no—
312Q8—21no—
313D4—21no—
314D4—21no—
315D4—21no—
316D4—42no—
317D4—81no—
318D4—3322no—
319D4—3322no—
320D4—6641no—
321D4—1662no—
322D4—1661no—
323D4—1661no—
324D4#1305058912yes—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—42no—
330C2×C4—831no—
331C2×C4—831no—
332C2×C4—831no—
333C2×C4—6641no—
334C2×C4—6641no—
335C2×C4—6641no—
336C2×C4—1661no—
337C2×C4—1661no—
338C2×C4—1661no—
339C2×C4—1661no—
340C2×C4—1661no—
341C2×C4D4×D8311yescharacteristic
342C2×C4C23×D8311yescharacteristic
343C8—21no—
344C8—21no—
345C8—1661no—
346C8—1661no—
347C22—21no—
348C22—22no—
349C22—21no—
350C22—21no—
351C22—41no—
352C22—41no—
353C22—41no—
354C22—41no—
355C22—41no—
356C22—3321no—
357C22—3321no—
358C22—3321no—
359C22—832no—
360C22—6644no—
361C22—1661no—
362C22—1661no—
363C22C2×D4×D8311yescharacteristic
364C4—21no—
365C4—41no—
366C4—41no—
367C4—3321no—
368C4—3321no—
369C4—3321no—
370C4—1661no—
371C4—1661no—
372C4—1661no—
373C4C2×#1305058911yescharacteristic
374C4#1458183711yescharacteristic
375C2—41no—
376C2—42no—
377C2—81no—
378C2—6641no—
379C2—1662no—
380C2#1912833811yescharacteristic
381C2#135×D8311yescharacteristic
382C2#1912850511yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#449×C83C2
socleC22×C83C22×D4
hypercenter#449D83
hypocenterC83#3297
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ #449

Complements

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

NH ≅ G/Ntype
1. #19128398C2semidirect
2. #19128339C2semidirect
3. #19128057C2semidirect
4. #19128035C2semidirect
5. #19128026C2semidirect
6. #19128003C2semidirect
7. #19127935C2semidirect
8. #19127927C2semidirect
9. #19127918C2semidirect
10. #19127833C2semidirect
11. D8×Dic83C2semidirect
12. (C22⋊D83)⋊C8C2semidirect
13. #19127697C2semidirect
14. #102×D83C2semidirect
15. #449×C83C2semidirect
18. C22×C83⋊C8C22semidirect
21. #14581783C22semidirect
32. D4×Dic83C22semidirect
35. #14581751C22semidirect
38. #14581726C22semidirect
39. #14581725C22semidirect
40. #14581724C22semidirect
41. Dic83⋊C8C22semidirect
42. #14581716C22semidirect
43. #14581716C22semidirect
48. #14581713C22semidirect
51. C8×Dic83C22semidirect
53. C2×SD16×C83C22semidirect
55. #121×C83C22semidirect
56. #103×C83C22semidirect
58. C22⋊C8×C83C22semidirect
192. C83#3297semidirect
202. #449D83semidirect
210. C2×SD16C2×D83semidirect
225. #121C2×D83semidirect
240. #103C2×D83semidirect
249. C22⋊C8C2×D83semidirect

Maximal subgroups

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

subgroupindex
#191283982
#191283392
#191280572
#191280352
#191280262
#191280032
#191279352
#191279272
#191279182
#191278332
D8×Dic832
(C22⋊D83)⋊C82
#191276972
#102×D832
#449×C832
#329783