Finite Group Databasebeta

← #24919227

Subgroups — #24919227

Order 10624.

464 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#19128512C211yesmaximal, characteristic, complemented
2#19128508C211yesmaximal, characteristic, complemented
3#19128504C211yesmaximal, characteristic, complemented
4#19128089C211yesmaximal, characteristic, complemented
5#19127969C211yesmaximal, characteristic, complemented
6#19127968C211yesmaximal, characteristic, complemented
7#19127964C211yesmaximal, characteristic, complemented
8#19127962C211yesmaximal, characteristic, complemented
9#19127879C211yesmaximal, characteristic, complemented
10#19127872C211yesmaximal, characteristic, complemented
11#19127869C211yesmaximal, characteristic, complemented
12#19127867C211yesmaximal, characteristic, complemented
13#19127823C211yesmaximal, characteristic, complemented
14#19127821C211yesmaximal, characteristic, complemented
15#576×C83C211yesmaximal, characteristic, complemented
16#14581839C2211yescharacteristic, complemented
17#14581838C2211yescharacteristic, complemented
18#14581824—21no—
19#14581822—21no—
20#14581822C2211yescharacteristic, complemented
21#14581822C2211yescharacteristic, complemented
22#14581818—21no—
23#14581818—21no—
24#14581818C2211yescharacteristic, complemented
25#14581818C2211yescharacteristic, complemented
26#14581817—21no—
27#14581817C2211yescharacteristic
28#14581817C2211yescharacteristic, complemented
29#14581816—21no—
30#14581813—21no—
31#14581812—21no—
32#14581812—21no—
33#14581812C2211yescharacteristic
34C2×#13050579C2211yescharacteristic, complemented
35C2×#13050578—21no—
36C2×#13050578C2211yescharacteristic, complemented
37SD16×D83—21no—
38C2×#13050575—21no—
39C2×#13050575C2211yescharacteristic, complemented
40#14581799C2211yescharacteristic, complemented
41#14581799C2211yescharacteristic, complemented
42#14581798C2211yescharacteristic, complemented
43#14581788C2211yescharacteristic, complemented
44#14581779C2211yescharacteristic, complemented
45#14581728C2211yescharacteristic, complemented
46#14581727C2211yescharacteristic, complemented
47#14581719C2211yescharacteristic, complemented
48#14581719C2211yescharacteristic, complemented
49#14581719C2211yescharacteristic, complemented
50#14581718C2211yescharacteristic, complemented
51#14581710C2211yescharacteristic, complemented
52#14581709C2211yescharacteristic, complemented
53#14581709C2211yescharacteristic, complemented
54#14581708C2211yescharacteristic, complemented
55#14581702C2211yescharacteristic, complemented
56H(5,𝔽2)×C83C2211yescharacteristic
57H(5,𝔽2)×C83C2211yescharacteristic, complemented
58AGL(1,ℤ/8)×C83—21no—
59AGL(1,ℤ/8)×C83—21no—
60AGL(1,ℤ/8)×C83C2211yescharacteristic
61AGL(1,ℤ/8)×C83C2211yescharacteristic, complemented
62#135×C83—21no—
63#135×C83C2211yescharacteristic
64C2×D8×C83—21no—
65C2×D8×C83C2211yescharacteristic, complemented
66#131×C83C2211yescharacteristic, complemented
67#13050591—21no—
68#13050590—21no—
69#13050590—21no—
70#13050590C2311yescharacteristic
71#13050589—21no—
72#13050589—41no—
73C2×Dic166—21no—
74C2×Dic166C2311yescharacteristic
75Q8×D83—21no—
76C2×D332—21no—
77C2×D332C2311yescharacteristic
78C2×C22⋊D83—41no—
79D4×D83—21no—
80D4×D83—41no—
81#13050580—21no—
82#13050580—21no—
83#13050580D411yescharacteristic, complemented
84#13050579—21no—
85#13050579—21no—
86#13050578—21no—
87#13050578—21no—
88#13050578—21no—
89#13050578—21no—
90#13050578—21no—
91#13050578—21no—
92#13050578D411yescharacteristic, complemented
93#13050577—21no—
94#13050577—21no—
95#13050577D411yescharacteristic, complemented
96#13050575—21no—
97#13050575—21no—
98#13050575—21no—
99#13050575—21no—
100#13050575—21no—
101#13050575—21no—
102#13050575D411yescharacteristic, complemented
103#13050574—21no—
104#13050574—21no—
105#13050574C2311yescharacteristic
106#13050574C2311yescharacteristic
107#13050574C2311yescharacteristic
108#13050573—21no—
109C2×C83⋊C8—21no—
110C2×C83⋊C8—21no—
111C2×C83⋊C8C2311yescharacteristic
112C8×D83—21no—
113C22⋊Dic83—41no—
114C4×Dic83C2311yescharacteristic
115#41×C83—21no—
116#41×C83—21no—
117#41×C83—21no—
118#41×C83—21no—
119#41×C83C2311yescharacteristic
120#41×C83C2311yescharacteristic
121#41×C83C2311yescharacteristic
122C2×D4×C83—21no—
123C2×D4×C83—21no—
124C2×D4×C83—21no—
125C2×D4×C83—21no—
126C2×D4×C83—41no—
127C2×D4×C83—41no—
128C2×D4×C83C2311yescharacteristic
129C2×D4×C83C2311yescharacteristic
130Q16×C83D411yescharacteristic, complemented
131SD16×C83—21no—
132SD16×C83—21no—
133SD16×C83D411yescharacteristic, complemented
134SD16×C83D411yescharacteristic, complemented
135D8×C83—21no—
136D8×C83—21no—
137D8×C83—21no—
138D8×C83—21no—
139D8×C83D411yescharacteristic, complemented
140M16×C83—21no—
141M16×C83C2311yescharacteristic
142C2×C8×C83—21no—
143C2×C8×C83C2311yescharacteristic
144C23×C83—21no—
145C23×C83—21no—
146C23×C83—41no—
147C23×C83—41no—
148C22×D83—21no—
149C22×D83—41no—
150Q8×C83C2×D411yescharacteristic
151Q8×C83C2×D411yescharacteristic
152Q8×C83C2×D411yescharacteristic
153D4×C83—21no—
154D4×C83—21no—
155D4×C83—21no—
156D4×C83—21no—
157D4×C83—21no—
158D4×C83—21no—
159D4×C83—21no—
160D4×C83—41no—
161D4×C83—41no—
162D4×C83—41no—
163D4×C83—41no—
164D4×C83C2×D411yescharacteristic
165D4×C83C2×D411yescharacteristic
166D4×C83C2×D411yescharacteristic
167C2×C4×C83—21no—
168C2×C4×C83—21no—
169C2×C4×C83—21no—
170C2×C4×C83—41no—
171C2×C4×C83—41no—
172C2×C4×C83C2411yescharacteristic
173C22⋊D83—21no—
174C22⋊D83—41no—
175C22⋊D83—41no—
176C22⋊D83—82no—
177C2×Dic83—21no—
178C2×Dic83—21no—
179C2×Dic83—41no—
180D332—21no—
181D332—21no—
182D332—41no—
183D332C2×D411yescharacteristic
184C4×D83—21no—
185C4×D83—41no—
186C4×D83—41no—
187Dic166—21no—
188Dic166—21no—
189Dic166—41no—
190Dic166C2×D411yescharacteristic
191C8×C83—21no—
192C8×C83C2×D411yescharacteristic
193C8×C83C2×D411yescharacteristic
194C83⋊C8—21no—
195C83⋊C8—21no—
196C83⋊C8—21no—
197C83⋊C8C2×D411yescharacteristic
198C83⋊C8C2×D411yescharacteristic
199C22×C83—21no—
200C22×C83—21no—
201C22×C83—21no—
202C22×C83—41no—
203C22×C83—41no—
204C22×C83—81no—
205C22×C83—81no—
206C22×C83—161no—
207C22×C83—161no—
208C22×C83H(5,𝔽2)11yescharacteristic
209C2×D83—21no—
210C2×D83—41no—
211C2×D83—81no—
212C2×D83—161no—
213C4×C83—21no—
214C4×C83—21no—
215C4×C83—21no—
216C4×C83C22×D411yescharacteristic
217C4×C83C22×D411yescharacteristic
218Dic83—21no—
219Dic83—41no—
220Dic83—41no—
221C2×C83—21no—
222C2×C83—41no—
223C2×C83—41no—
224C2×C83—41no—
225C2×C83—81no—
226C2×C83—81no—
227C2×C83D4211yescharacteristic
228D83—41no—
229D83—81no—
230#3288—831nomaximal, Sylow
231C83#328811yescharacteristic, complemented, Sylow
232#577—831no—
233#577—831no—
234#576—831no—
235#576D8311yescharacteristic, complemented
236#496—831no—
237#472—831no—
238#472—831no—
239#471—831no—
240#471—831no—
241#455—831no—
242#454—831no—
243#453—831no—
244#453—831no—
245#444—831no—
246#444—831no—
2472−1+4—831no—
248H(5,𝔽2)—831no—
249H(5,𝔽2)C2×D8311yescharacteristic
250H(5,𝔽2)C2×D8311yescharacteristic, complemented
251#137—831no—
252#137—831no—
253#137—1661no—
254#137—1661no—
255AGL(1,ℤ/8)—21no—
256AGL(1,ℤ/8)—21no—
257AGL(1,ℤ/8)—831no—
258AGL(1,ℤ/8)—831no—
259AGL(1,ℤ/8)—831no—
260AGL(1,ℤ/8)—831no—
261AGL(1,ℤ/8)—1661no—
262AGL(1,ℤ/8)—1661no—
263AGL(1,ℤ/8)—1661no—
264AGL(1,ℤ/8)—1661no—
265AGL(1,ℤ/8)C2×D8311yescharacteristic
266AGL(1,ℤ/8)C2×D8311yescharacteristic, complemented
267#135—21no—
268#135—831no—
269#135—1661no—
270#135—1661no—
271#135—1661no—
272#135C2×D8311yescharacteristic
273C2×SD16—831no—
274C2×SD16—831no—
275C2×SD16—1661no—
276C2×SD16—1661no—
277C2×D8—21no—
278C2×D8—831no—
279C2×D8—1661no—
280C2×D8C2×D8311yescharacteristic, complemented
281#131—831no—
282#131—831no—
283#131—831no—
284#131C2×D8311yescharacteristic, complemented
285#127—831no—
286#124—831no—
287#108—831no—
288#108—831no—
289#104—831no—
290#104—831no—
291#104—831no—
292#104—831no—
293#101—831no—
294#100—831no—
295#100—831no—
296#100—831no—
297#97—831no—
298#41—21no—
299#41—21no—
300#41—21no—
301#41—21no—
302#41—3321no—
303#41—831no—
304#41—1661no—
305#41—1661no—
306#41—1661no—
307#41—1661no—
308#41C22×D8311yescharacteristic
309#41C22×D8311yescharacteristic
310#41C22×D8311yescharacteristic
311C2×Q8—831no—
312C2×Q8—1661no—
313C2×Q8—1661no—
314C2×D4—21no—
315C2×D4—21no—
316C2×D4—21no—
317C2×D4—21no—
318C2×D4—41no—
319C2×D4—41no—
320C2×D4—3321no—
321C2×D4—3321no—
322C2×D4—831no—
323C2×D4—1661no—
324C2×D4—1661no—
325C2×D4C22×D8311yescharacteristic
326C2×D4C22×D8311yescharacteristic
327Q16—831no—
328Q16—1661no—
329Q16—1661no—
330Q16C22⋊D8311yescharacteristic, complemented
331SD16—21no—
332SD16—21no—
333SD16—831no—
334SD16—831no—
335SD16—1661no—
336SD16—1661no—
337SD16—1661no—
338SD16—1661no—
339SD16—1661no—
340SD16—1661no—
341SD16—1661no—
342SD16—1661no—
343SD16—1661no—
344SD16—1661no—
345SD16C22⋊D8311yescharacteristic, complemented
346SD16C22⋊D8311yescharacteristic, complemented
347D8—21no—
348D8—21no—
349D8—21no—
350D8—21no—
351D8—831no—
352D8—1661no—
353D8—1661no—
354D8—1661no—
355D8—1661no—
356D8—1661no—
357D8—1661no—
358D8C22⋊D8311yescharacteristic, complemented
359M16—21no—
360M16—831no—
361M16—831no—
362M16—831no—
363M16—1661no—
364M16—1661no—
365M16—1661no—
366M16C22×D8311yescharacteristic
367C2×C8—21no—
368C2×C8—831no—
369C2×C8—1661no—
370C2×C8—1661no—
371C2×C8—1661no—
372C2×C8C22×D8311yescharacteristic
373C22⋊C4—3321no—
374C42—831no—
375C23—21no—
376C23—21no—
377C23—41no—
378C23—41no—
379C23—3321no—
380C23—1661no—
381Q8—3321no—
382Q8—831no—
383Q8—1661no—
384Q8—1661no—
385Q8D4×D8311yescharacteristic
386Q8C2×C22⋊D8311yescharacteristic
387Q8C2×C22⋊D8311yescharacteristic
388D4—21no—
389D4—21no—
390D4—21no—
391D4—21no—
392D4—21no—
393D4—21no—
394D4—21no—
395D4—41no—
396D4—41no—
397D4—41no—
398D4—41no—
399D4—3321no—
400D4—3321no—
401D4—3321no—
402D4—831no—
403D4—6642no—
404D4—1661no—
405D4—1661no—
406D4—1661no—
407D4D4×D8311yescharacteristic
408D4C2×C22⋊D8311yescharacteristic
409D4C2×C22⋊D8311yescharacteristic
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—3321no—
416C2×C4—3321no—
417C2×C4—3321no—
418C2×C4—1661no—
419C2×C4—1661no—
420C2×C4—1661no—
421C2×C4C23×D8311yescharacteristic
422C8—21no—
423C8—831no—
424C8—831no—
425C8—1661no—
426C8—1661no—
427C8—1661no—
428C8C2×C22⋊D8311yescharacteristic
429C8C2×C22⋊D8311yescharacteristic
430C22—21no—
431C22—21no—
432C22—21no—
433C22—41no—
434C22—41no—
435C22—81no—
436C22—81no—
437C22—161no—
438C22—161no—
439C22—13281no—
440C22—3321no—
441C22—6641no—
442C22—1661no—
443C22#1458183711yescharacteristic
444C4—21no—
445C4—21no—
446C4—21no—
447C4—3321no—
448C4—3321no—
449C4—1661no—
450C4C2×D4×D8311yescharacteristic
451C4C22×C22⋊D8311yescharacteristic
452C2—21no—
453C2—41no—
454C2—41no—
455C2—41no—
456C2—81no—
457C2—81no—
458C2—3321no—
459C2—6641no—
460C2D4×C22⋊D8311yescharacteristic
subgroupG/H
centerC2D4×C22⋊D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#576×C83C2
socleC2×C83D42
hypercenter#576D83
hypocenterC83#3288
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C2 ◁ C2×C4 ◁ #576

Complements

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

NH ≅ G/Ntype
1. #19128512C2semidirect
2. #19128508C2semidirect
3. #19128504C2semidirect
4. #19128089C2semidirect
5. #19127969C2semidirect
6. #19127968C2semidirect
7. #19127964C2semidirect
8. #19127962C2semidirect
9. #19127879C2semidirect
10. #19127872C2semidirect
11. #19127869C2semidirect
12. #19127867C2semidirect
13. #19127823C2semidirect
14. #19127821C2semidirect
15. #576×C83C2semidirect
16. #14581839C22semidirect
17. #14581838C22semidirect
20. #14581822C22semidirect
21. #14581822C22semidirect
24. #14581818C22semidirect
25. #14581818C22semidirect
28. #14581817C22semidirect
34. C2×#13050579C22semidirect
36. C2×#13050578C22semidirect
39. C2×#13050575C22semidirect
40. #14581799C22semidirect
41. #14581799C22semidirect
42. #14581798C22semidirect
43. #14581788C22semidirect
44. #14581779C22semidirect
45. #14581728C22semidirect
46. #14581727C22semidirect
47. #14581719C22semidirect
48. #14581719C22semidirect
49. #14581719C22semidirect
50. #14581718C22semidirect
51. #14581710C22semidirect
52. #14581709C22semidirect
53. #14581709C22semidirect
54. #14581708C22semidirect
55. #14581702C22semidirect
57. H(5,𝔽2)×C83C22semidirect
61. AGL(1,ℤ/8)×C83C22semidirect
65. C2×D8×C83C22semidirect
66. #131×C83C22semidirect
83. #13050580D4semidirect
92. #13050578D4semidirect
95. #13050577D4semidirect
102. #13050575D4semidirect
130. Q16×C83D4semidirect
133. SD16×C83D4semidirect
134. SD16×C83D4semidirect
139. D8×C83D4semidirect
231. C83#3288semidirect
235. #576D83semidirect
250. H(5,𝔽2)C2×D83semidirect
266. AGL(1,ℤ/8)C2×D83semidirect
280. C2×D8C2×D83semidirect
284. #131C2×D83semidirect
330. Q16C22⋊D83semidirect
345. SD16C22⋊D83semidirect
346. SD16C22⋊D83semidirect
358. D8C22⋊D83semidirect

Maximal subgroups

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

subgroupindex
#191285122
#191285082
#191285042
#191280892
#191279692
#191279682
#191279642
#191279622
#191278792
#191278722
#191278692
#191278672
#191278232
#191278212
#576×C832
#328883