Finite Group Databasebeta

← #24919665

Subgroups — #24919665

Order 10624.

414 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#19128399C211yesmaximal, characteristic, complemented
2#19128339C211yesmaximal, characteristic, complemented
3#19128057C211yesmaximal, characteristic, complemented
4#19128040C211yesmaximal, characteristic, complemented
5#19128014C211yesmaximal, characteristic, complemented
6#19128003C211yesmaximal, characteristic, complemented
7#19127954C211yesmaximal, characteristic, complemented
8#19127950C211yesmaximal, characteristic, complemented
9#19127919C211yesmaximal, characteristic, complemented
10#19127846C211yesmaximal, characteristic, complemented
11#19127836C211yesmaximal, characteristic, complemented
12#19127781C211yesmaximal, characteristic, complemented
13#19127777C211yesmaximal, characteristic, complemented
14#19127704C211yesmaximal, characteristic, complemented
15#417×C83C211yesmaximal, characteristic, complemented
16C2×D4×D83C2211yescharacteristic
17C2×#13050577C2211yescharacteristic, complemented
18C2×D664C2211yescharacteristic
19C2×#13050575C2211yescharacteristic
20C2×#13050573C2211yescharacteristic
21#14581787—21no—
22#14581783C2211yescharacteristic, complemented
23#14581781C2211yescharacteristic
24#14581777—21no—
25#14581775—21no—
26#14581768C2211yescharacteristic
27#14581767C2211yescharacteristic
28#14581766—21no—
29#14581766C2211yescharacteristic
30#14581766C2211yescharacteristic, complemented
31#14581764—21no—
32#14581760—21no—
33D4×Dic83C2211yescharacteristic
34#14581754C2211yescharacteristic, complemented
35C4×C22⋊D83—21no—
36C4×C22⋊D83C2211yescharacteristic
37#14581750C2211yescharacteristic
38C2×(C2×D83)⋊C4—21no—
39#14581725C2211yescharacteristic, complemented
40#14581725C2211yescharacteristic, complemented
41#14581724C2211yescharacteristic, complemented
42Dic83⋊C8C2211yescharacteristic, complemented
43#14581716C2211yescharacteristic, complemented
44#14581714C2211yescharacteristic
45#14581714C2211yescharacteristic, complemented
46#14581713C2211yescharacteristic
47#14581712C2211yescharacteristic
48#14581712C2211yescharacteristic
49C22⋊(C83⋊C8)C2211yescharacteristic, complemented
50(C2×D83)⋊C8C2211yescharacteristic
51C8×Dic83C2211yescharacteristic, complemented
52C2×#41×C83C2211yescharacteristic
53C2×M16×C83C2211yescharacteristic, complemented
54C2×C4⋊C4×C83C2211yescharacteristic, complemented
55#103×C83C2211yescharacteristic, complemented
56#103×C83C2211yescharacteristic, complemented
57#102×C83C2211yescharacteristic
58#102×C83C2211yescharacteristic
59C23×D83—21no—
60C2×D332—21no—
61C2×D332C2311yescharacteristic
62C2×D332C2311yescharacteristic
63C2×C22⋊D83—21no—
64C2×C22⋊D83—21no—
65C2×C22⋊D83—21no—
66C2×C22⋊D83D411yescharacteristic
67D4×D83—22no—
68D4×D83—41no—
69C22×Dic83—21no—
70C2×C4×D83—21no—
71C2×C4×D83C2311yescharacteristic
72#13050577—41no—
73D664—41no—
74#13050575—22no—
75#13050573—41no—
76C2×C83⋊C8C2311yescharacteristic
77C2×C83⋊C8C2311yescharacteristic
78C4⋊Dic83—21no—
79C4⋊Dic83C2311yescharacteristic
80C4⋊Dic83C2311yescharacteristic
81Dic83⋊C4—21no—
82Dic83⋊C4—21no—
83Dic83⋊C4D411yescharacteristic
84C22⋊Dic83—21no—
85C22⋊Dic83D411yescharacteristic
86(C2×D83)⋊C4—21no—
87(C2×D83)⋊C4—21no—
88(C2×D83)⋊C4—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)⋊C4D411yescharacteristic
95C4×Dic83—21no—
96C4×Dic83C2311yescharacteristic
97#41×C83—22no—
98#41×C83—41no—
99C2×Q8×C83C2311yescharacteristic
100C2×D4×C83—21no—
101C2×D4×C83C2311yescharacteristic
102C22×C4×C83—21no—
103C22×C4×C83—21no—
104C22×C4×C83C2311yescharacteristic
105M16×C83—41no—
106C2×C8×C83C2311yescharacteristic
107C2×C8×C83C2311yescharacteristic
108C4⋊C4×C83—21no—
109C4⋊C4×C83C2311yescharacteristic
110C4⋊C4×C83C2311yescharacteristic
111C23×C83—21no—
112C23×C83C2×D411yescharacteristic
113C22×D83—21no—
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—42no—
123D4×C83#4112yes—
124C2×C4×C83—21no—
125C2×C4×C83—21no—
126C2×C4×C83—21no—
127C2×C4×C83—21no—
128C2×C4×C83—21no—
129C2×C4×C83—42no—
130C2×C4×C83—81no—
131C2×C4×C83C2×D411yescharacteristic
132C2×C4×C83C2411yescharacteristic
133C22⋊D83—41no—
134C22⋊D83—42no—
135C22⋊D83—42no—
136C22⋊D83—81no—
137C2×Dic83—21no—
138C2×Dic83—21no—
139C2×Dic83—81no—
140C2×Dic83C2×D411yescharacteristic
141C2×Dic83C2×D411yescharacteristic
142C2×Dic83C2×D411yescharacteristic
143D332—21no—
144D332—21no—
145D332—21no—
146D332—21no—
147D332—81no—
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×C83C22×D411yescharacteristic
162C2×D83—21no—
163C2×D83—21no—
164C2×D83—21no—
165C2×D83—41no—
166C2×D83—41no—
167C2×D83—41no—
168C2×D83—41no—
169C2×D83—41no—
170C2×D83—41no—
171C2×D83—84no—
172C4×C83—21no—
173C4×C83—41no—
174C4×C83—41no—
175C4×C83—41no—
176C4×C83C2×#4111yescharacteristic
177C4×C83H(5,𝔽2)11yescharacteristic
178Dic83—21no—
179Dic83—21no—
180Dic83—21no—
181Dic83—41no—
182Dic83—41no—
183C2×C83—42no—
184C2×C83—41no—
185C2×C83#54611yescharacteristic
186C2×C83#57511yescharacteristic
187C2×C83#57611yescharacteristic
188D83—41no—
189D83—81no—
190D83—81no—
191#3322—831nomaximal, Sylow
192C83#332211yescharacteristic, complemented, Sylow
193#553—831no—
194#546—831no—
195#488—831no—
196#485—831no—
197#480—831no—
198#478—831no—
199#469—831no—
200#468—831no—
201#463—831no—
202#449—831no—
203#447—831no—
204C4×D8—831no—
205#436—831no—
206#418—831no—
207#417D8311yescharacteristic, complemented
208C2×#41C2×D8311yescharacteristic
209C22×D4—831no—
210C2×SD16—831no—
211C2×D8—831no—
212C2×D8—831no—
213C2×M16—831no—
214C2×M16C2×D8311yescharacteristic, complemented
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—831no—
224#121—1661no—
225#121—1661no—
226#120—1661no—
227C4×D4—831no—
228C4×D4—831no—
229C4×D4—831no—
230C4×D4—1661no—
231#117—831no—
232C2×C4⋊C4C2×D8311yescharacteristic, complemented
233C2×C22⋊C4—1661no—
234#107—831no—
235#107—831no—
236#106—831no—
237C4⋊C8—831no—
238#103—831no—
239#103C2×D8311yescharacteristic, complemented
240#103C2×D8311yescharacteristic, complemented
241#102—831no—
242#102—831no—
243#102—831no—
244#102—831no—
245#102—831no—
246#102C2×D8311yescharacteristic
247#102C2×D8311yescharacteristic
248C22⋊C8—831no—
249C22⋊C8—831no—
250C4×C8—831no—
251C24—1661no—
252#41—22no—
253#41—41no—
254C2×Q8C22×D8311yescharacteristic
255C2×D4—21no—
256C2×D4—3321no—
257C2×D4—831no—
258C2×D4—831no—
259C2×D4—831no—
260C2×D4—1661no—
261C2×D4—1661no—
262C2×D4—1661no—
263C2×D4—1662no—
264C2×D4—1661no—
265C2×D4C22×D8311yescharacteristic
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—831no—
269C22×C4—1661no—
270C22×C4—1661no—
271C22×C4C22×D8311yescharacteristic
272SD16—3321no—
273D8—3321no—
274D8—1662no—
275M16—41no—
276M16—3321no—
277C2×C8—831no—
278C2×C8—831no—
279C2×C8C22×D8311yescharacteristic
280C2×C8C22×D8311yescharacteristic
281C4⋊C4—21no—
282C4⋊C4—831no—
283C4⋊C4—831no—
284C4⋊C4—831no—
285C4⋊C4—1661no—
286C4⋊C4—1661no—
287C4⋊C4—1661no—
288C4⋊C4C22×D8311yescharacteristic
289C4⋊C4C22×D8311yescharacteristic
290C22⋊C4—831no—
291C22⋊C4—831no—
292C22⋊C4—1661no—
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—3322no—
305C23—3322no—
306C23—831no—
307C23—6641no—
308C23—1661no—
309C23—1661no—
310C23D4×D8311yescharacteristic
311Q8—21no—
312Q8—21no—
313D4—21no—
314D4—42no—
315D4—3322no—
316D4—3322no—
317D4—3321no—
318D4—6641no—
319D4—6641no—
320D4—1661no—
321D4—1661no—
322D4—1661no—
323D4—1661no—
324D4#1305059012yes—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—42no—
331C2×C4—81no—
332C2×C4—831no—
333C2×C4—831no—
334C2×C4—831no—
335C2×C4—6641no—
336C2×C4—6641no—
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—3321no—
354C22—3321no—
355C22—3321no—
356C22—3321no—
357C22—3321no—
358C22—3321no—
359C22—6644no—
360C22—1661no—
361C22—1661no—
362C22—1661no—
363C22C2×D4×D8311yescharacteristic
364C4—21no—
365C4—41no—
366C4—41no—
367C4—41no—
368C4—3321no—
369C4—3321no—
370C4—1661no—
371C4—1661no—
372C4—1661no—
373C4C2×#1305059011yescharacteristic
374C4#1458183711yescharacteristic
375C2—41no—
376C2—42no—
377C2—3321no—
378C2—6641no—
379C2—6641no—
380C2#1912834111yescharacteristic
381C2#1912850011yescharacteristic
382C2#1912850511yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#417×C83C2
socleC22×C83C22×D4
hypercenter#417D83
hypocenterC83#3322
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ #417

Complements

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

NH ≅ G/Ntype
1. #19128399C2semidirect
2. #19128339C2semidirect
3. #19128057C2semidirect
4. #19128040C2semidirect
5. #19128014C2semidirect
6. #19128003C2semidirect
7. #19127954C2semidirect
8. #19127950C2semidirect
9. #19127919C2semidirect
10. #19127846C2semidirect
11. #19127836C2semidirect
12. #19127781C2semidirect
13. #19127777C2semidirect
14. #19127704C2semidirect
15. #417×C83C2semidirect
17. C2×#13050577C22semidirect
22. #14581783C22semidirect
30. #14581766C22semidirect
34. #14581754C22semidirect
39. #14581725C22semidirect
40. #14581725C22semidirect
41. #14581724C22semidirect
42. Dic83⋊C8C22semidirect
43. #14581716C22semidirect
45. #14581714C22semidirect
49. C22⋊(C83⋊C8)C22semidirect
51. C8×Dic83C22semidirect
53. C2×M16×C83C22semidirect
54. C2×C4⋊C4×C83C22semidirect
55. #103×C83C22semidirect
56. #103×C83C22semidirect
192. C83#3322semidirect
207. #417D83semidirect
214. C2×M16C2×D83semidirect
232. C2×C4⋊C4C2×D83semidirect
239. #103C2×D83semidirect
240. #103C2×D83semidirect

Maximal subgroups

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

subgroupindex
#191283992
#191283392
#191280572
#191280402
#191280142
#191280032
#191279542
#191279502
#191279192
#191278462
#191278362
#191277812
#191277772
#191277042
#417×C832
#332283