Finite Group Databasebeta

← #24919255

Subgroups — #24919255

Order 10624.

460 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#19128513C211yesmaximal, characteristic, complemented
2#19128511C211yesmaximal, characteristic, complemented
3#19128510C211yesmaximal, characteristic, complemented
4#19128507C211yesmaximal, characteristic, complemented
5#19128072C211yesmaximal, characteristic, complemented
6#19127967C211yesmaximal, characteristic, complemented
7#19127965C211yesmaximal, characteristic, complemented
8#19127963C211yesmaximal, characteristic, complemented
9#19127886C211yesmaximal, characteristic, complemented
10#19127882C211yesmaximal, characteristic, complemented
11#19127871C211yesmaximal, characteristic, complemented
12#19127867C211yesmaximal, characteristic, complemented
13#19127817C211yesmaximal, characteristic, complemented
14#19127816C211yesmaximal, characteristic, complemented
15#471×C83C211yesmaximal, characteristic, complemented
16#14581840C2211yescharacteristic
17#14581839C2211yescharacteristic
18#14581838C2211yescharacteristic, complemented
19#14581837C2211yescharacteristic, complemented
20#14581825—21no—
21#14581825C2211yescharacteristic, complemented
22#14581825C2211yescharacteristic, complemented
23#14581824—21no—
24#14581823C2211yescharacteristic, complemented
25#14581821—21no—
26#14581821—21no—
27#14581820—21no—
28#14581820C2211yescharacteristic
29#14581820C2211yescharacteristic
30#14581819—21no—
31#14581819—21no—
32#14581818C2211yescharacteristic
33#14581816—21no—
34#14581815—21no—
35#14581814—21no—
36#14581814C2211yescharacteristic
37#14581813—21no—
38#14581812C2211yescharacteristic
39#14581811—21no—
40C2×#13050579—21no—
41C2×#13050579C2211yescharacteristic, complemented
42C2×#13050578C2211yescharacteristic, complemented
43C2×#13050577C2211yescharacteristic, complemented
44SD16×D83—21no—
45SD16×D83—21no—
46SD16×D83—21no—
47#14581800C2211yescharacteristic, complemented
48#14581800C2211yescharacteristic, complemented
49#14581798C2211yescharacteristic, complemented
50#14581790C2211yescharacteristic, complemented
51#14581788C2211yescharacteristic
52#14581727C2211yescharacteristic, complemented
53#14581719C2211yescharacteristic, complemented
54#14581718C2211yescharacteristic, complemented
55#14581718C2211yescharacteristic, complemented
56#14581718C2211yescharacteristic, complemented
57#14581710C2211yescharacteristic, complemented
58#14581708C2211yescharacteristic, complemented
59C8×Dic83C2211yescharacteristic, complemented
60#137×C83C2211yescharacteristic, complemented
61AGL(1,ℤ/8)×C83C2211yescharacteristic
62C2×SD16×C83C2211yescharacteristic, complemented
63#131×C83C2211yescharacteristic, complemented
64#108×C83C2211yescharacteristic, complemented
65#101×C83C2211yescharacteristic, complemented
66#100×C83C2211yescharacteristic, complemented
67#13050591—21no—
68#13050591—21no—
69#13050591—21no—
70#13050590—21no—
71#13050590—21no—
72#13050590—41no—
73#13050590C2311yescharacteristic
74#13050590C2311yescharacteristic
75#13050590C2311yescharacteristic
76#13050589—21no—
77#13050589—21no—
78#13050589—21no—
79#13050589—41no—
80C2×Dic166—21no—
81C2×Dic166C2311yescharacteristic
82Q8×D83—21no—
83Q8×D83—21no—
84Q8×D83—21no—
85C2×D332—21no—
86C2×D332C2311yescharacteristic
87C2×C22⋊D83—41no—
88C2×C22⋊D83—41no—
89D4×D83—21no—
90D4×D83—21no—
91D4×D83—21no—
92D4×D83—41no—
93Dic332—21no—
94Dic332—21no—
95Dic332D411yescharacteristic, complemented
96#13050580—21no—
97#13050580D411yescharacteristic, complemented
98#13050579—21no—
99#13050579—21no—
100#13050579—21no—
101#13050579—21no—
102#13050579—21no—
103#13050579—21no—
104#13050579D411yescharacteristic, complemented
105#13050579D411yescharacteristic, complemented
106#13050578—21no—
107#13050578—21no—
108#13050578—21no—
109#13050578D411yescharacteristic, complemented
110#13050577—21no—
111#13050577—21no—
112#13050577—21no—
113#13050577D411yescharacteristic, complemented
114D664—21no—
115D664—21no—
116D664D411yescharacteristic, complemented
117#13050575—21no—
118#13050575D411yescharacteristic, complemented
119#13050574C2311yescharacteristic
120#13050573—21no—
121#13050573—21no—
122#13050573—21no—
123C2×C83⋊C8C2311yescharacteristic
124C8×D83—21no—
125C8×D83—21no—
126C8×D83—21no—
127Dic83⋊C4—41no—
128C4×Dic83C2311yescharacteristic
129#41×C83C2311yescharacteristic
130C2×Q8×C83C2311yescharacteristic
131C2×D4×C83C2311yescharacteristic
132Q16×C83—21no—
133SD16×C83—21no—
134SD16×C83—21no—
135SD16×C83—21no—
136SD16×C83—21no—
137D8×C83—21no—
138M16×C83—21no—
139M16×C83C2311yescharacteristic
140M16×C83C2311yescharacteristic
141M16×C83C2311yescharacteristic
142C2×C8×C83—21no—
143C2×C8×C83C2311yescharacteristic
144C23×C83—21no—
145C22×D83—21no—
146C22×D83—41no—
147C22×D83—41no—
148Q8×C83—21no—
149Q8×C83—21no—
150Q8×C83C2×D411yescharacteristic
151D4×C83—21no—
152D4×C83—21no—
153D4×C83—21no—
154D4×C83C2×D411yescharacteristic
155C2×C4×C83—21no—
156C2×C4×C83—21no—
157C2×C4×C83C2411yescharacteristic
158C22⋊D83—21no—
159C22⋊D83—21no—
160C22⋊D83—21no—
161C22⋊D83—41no—
162C22⋊D83—41no—
163C22⋊D83—41no—
164C22⋊D83—41no—
165C22⋊D83—82no—
166C2×Dic83—21no—
167C2×Dic83—21no—
168C2×Dic83—41no—
169C2×Dic83—41no—
170D332—21no—
171D332—21no—
172D332—41no—
173D332—41no—
174D332C2×D411yescharacteristic
175D332C2×D411yescharacteristic
176D332C2×D411yescharacteristic
177C4×D83—21no—
178C4×D83—21no—
179C4×D83—21no—
180C4×D83—41no—
181C4×D83—41no—
182C4×D83—41no—
183C4×D83—41no—
184Dic166—21no—
185Dic166—21no—
186Dic166—41no—
187Dic166—41no—
188Dic166C2×D411yescharacteristic
189Dic166C2×D411yescharacteristic
190Dic166C2×D411yescharacteristic
191C8×C83—21no—
192C8×C83—21no—
193C8×C83—21no—
194C8×C83C2×D411yescharacteristic
195C8×C83C2×D411yescharacteristic
196C83⋊C8—21no—
197C83⋊C8C2×D411yescharacteristic
198C83⋊C8C2×D411yescharacteristic
199C22×C83—21no—
200C22×C83—41no—
201C22×C83—81no—
202C22×C83H(5,𝔽2)11yescharacteristic
203C2×D83—21no—
204C2×D83—21no—
205C2×D83—21no—
206C2×D83—41no—
207C2×D83—81no—
208C2×D83—161no—
209C2×D83—161no—
210C4×C83—21no—
211C4×C83—41no—
212C4×C83C22×D411yescharacteristic
213C4×C83C22×D411yescharacteristic
214Dic83—22no—
215Dic83—21no—
216Dic83—21no—
217Dic83—21no—
218Dic83—41no—
219C2×C83—21no—
220C2×C83—41no—
221C2×C83—81no—
222C2×C83D4211yescharacteristic
223D83—41no—
224D83—41no—
225D83—41no—
226D83—81no—
227#3291—831nomaximal, Sylow
228C83#329111yescharacteristic, complemented, Sylow
229#577—831no—
230#577—831no—
231#577—831no—
232#577—831no—
233#492—831no—
234#471—831no—
235#471—831no—
236#471—831no—
237#471D8311yescharacteristic, complemented
238#456—831no—
239#456—831no—
240#453—831no—
241#453—831no—
242#443—831no—
243#443—831no—
2442−1+4—831no—
2452−1+4—831no—
246H(5,𝔽2)—831no—
247H(5,𝔽2)—831no—
248#137—831no—
249#137—831no—
250#137—831no—
251#137—1661no—
252#137—1661no—
253#137—1661no—
254#137—1661no—
255#137C2×D8311yescharacteristic, complemented
256AGL(1,ℤ/8)—831no—
257AGL(1,ℤ/8)—831no—
258AGL(1,ℤ/8)—831no—
259AGL(1,ℤ/8)—1661no—
260AGL(1,ℤ/8)—1661no—
261AGL(1,ℤ/8)—1661no—
262AGL(1,ℤ/8)—1661no—
263AGL(1,ℤ/8)C2×D8311yescharacteristic
264#135—831no—
265#135—831no—
266#135—1661no—
267#135—1661no—
268#135—1661no—
269#135—1661no—
270C2×SD16—831no—
271C2×SD16—831no—
272C2×SD16—831no—
273C2×SD16—1661no—
274C2×SD16—1661no—
275C2×SD16—1661no—
276C2×SD16—1661no—
277C2×SD16C2×D8311yescharacteristic, complemented
278#131—831no—
279#131—831no—
280#131—831no—
281#131C2×D8311yescharacteristic, complemented
282C4⋊Q8—831no—
283#127—831no—
284#108—831no—
285#108C2×D8311yescharacteristic, complemented
286#104—831no—
287#104—831no—
288#104—831no—
289#104—831no—
290#101—831no—
291#101C2×D8311yescharacteristic, complemented
292#100—831no—
293#100C2×D8311yescharacteristic, complemented
294C4×C8—831no—
295#41—3321no—
296#41—3321no—
297#41—831no—
298#41—831no—
299#41—831no—
300#41—1661no—
301#41—1661no—
302#41—1661no—
303#41—1661no—
304#41—1661no—
305#41—1661no—
306#41—1661no—
307#41—1661no—
308#41C22×D8311yescharacteristic
309C2×Q8—831no—
310C2×Q8—1661no—
311C2×Q8—1661no—
312C2×Q8—1661no—
313C2×Q8—1661no—
314C2×Q8C22×D8311yescharacteristic
315C2×D4—3321no—
316C2×D4—3321no—
317C2×D4—3321no—
318C2×D4—831no—
319C2×D4—1661no—
320C2×D4—1661no—
321C2×D4—1661no—
322C2×D4—1661no—
323C2×D4C22×D8311yescharacteristic
324Q16—21no—
325Q16—831no—
326Q16—831no—
327Q16—1661no—
328Q16—1661no—
329Q16—1661no—
330SD16—21no—
331SD16—21no—
332SD16—21no—
333SD16—21no—
334SD16—831no—
335SD16—831no—
336SD16—831no—
337SD16—831no—
338SD16—1661no—
339SD16—1661no—
340SD16—1661no—
341SD16—1661no—
342SD16—1661no—
343SD16—1661no—
344SD16—1661no—
345SD16—1661no—
346SD16—1661no—
347SD16—1661no—
348SD16—1661no—
349SD16—1661no—
350D8—21no—
351D8—831no—
352D8—831no—
353D8—1661no—
354D8—1661no—
355D8—1661no—
356M16—21no—
357M16—831no—
358M16—1661no—
359M16—1661no—
360M16—1661no—
361M16C22×D8311yescharacteristic
362M16C22×D8311yescharacteristic
363M16C22×D8311yescharacteristic
364C2×C8—21no—
365C2×C8—831no—
366C2×C8—1661no—
367C2×C8—1661no—
368C2×C8—1661no—
369C2×C8C22×D8311yescharacteristic
370C4⋊C4—3321no—
371C42—831no—
372C23—21no—
373C23—3321no—
374C23—3321no—
375C23—1661no—
376Q8—21no—
377Q8—21no—
378Q8—3321no—
379Q8—3321no—
380Q8—831no—
381Q8—831no—
382Q8—831no—
383Q8—1661no—
384Q8—1661no—
385Q8D4×D8311yescharacteristic
386D4—21no—
387D4—21no—
388D4—21no—
389D4—3321no—
390D4—3321no—
391D4—3321no—
392D4—3321no—
393D4—3321no—
394D4—3321no—
395D4—831no—
396D4—831no—
397D4—831no—
398D4—6642no—
399D4—1661no—
400D4—1661no—
401D4—1661no—
402D4—1661no—
403D4—1661no—
404D4D4×D8311yescharacteristic
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—3321no—
408C2×C4—3321no—
409C2×C4—3321no—
410C2×C4—3321no—
411C2×C4—3321no—
412C2×C4—3321no—
413C2×C4—1661no—
414C2×C4—1661no—
415C2×C4—1661no—
416C2×C4—1661no—
417C2×C4—1661no—
418C2×C4C23×D8311yescharacteristic
419C8—21no—
420C8—21no—
421C8—21no—
422C8—831no—
423C8—831no—
424C8—1661no—
425C8D4×D8311yescharacteristic
426C8D4×D8311yescharacteristic
427C22—21no—
428C22—41no—
429C22—81no—
430C22—13281no—
431C22—13281no—
432C22—3321no—
433C22—6641no—
434C22—1661no—
435C22—1661no—
436C22—1661no—
437C22#1458183811yescharacteristic
438C4—21no—
439C4—41no—
440C4—3321no—
441C4—1662no—
442C4—1661no—
443C4—1661no—
444C4—1661no—
445C4C2×D4×D8311yescharacteristic
446C4C2×D4×D8311yescharacteristic
447C2—21no—
448C2—41no—
449C2—81no—
450C2—3321no—
451C2—3321no—
452C2—3321no—
453C2—6641no—
454C2#1912833511yescharacteristic
subgroupG/H
centerC2#19128335
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#471×C83C2
socleC2×C83D42
hypercenter#471D83
hypocenterC83#3291
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C2 ◁ C2×C4 ◁ #471

Complements

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

NH ≅ G/Ntype
1. #19128513C2semidirect
2. #19128511C2semidirect
3. #19128510C2semidirect
4. #19128507C2semidirect
5. #19128072C2semidirect
6. #19127967C2semidirect
7. #19127965C2semidirect
8. #19127963C2semidirect
9. #19127886C2semidirect
10. #19127882C2semidirect
11. #19127871C2semidirect
12. #19127867C2semidirect
13. #19127817C2semidirect
14. #19127816C2semidirect
15. #471×C83C2semidirect
18. #14581838C22semidirect
19. #14581837C22semidirect
21. #14581825C22semidirect
22. #14581825C22semidirect
24. #14581823C22semidirect
41. C2×#13050579C22semidirect
42. C2×#13050578C22semidirect
43. C2×#13050577C22semidirect
47. #14581800C22semidirect
48. #14581800C22semidirect
49. #14581798C22semidirect
50. #14581790C22semidirect
52. #14581727C22semidirect
53. #14581719C22semidirect
54. #14581718C22semidirect
55. #14581718C22semidirect
56. #14581718C22semidirect
57. #14581710C22semidirect
58. #14581708C22semidirect
59. C8×Dic83C22semidirect
60. #137×C83C22semidirect
62. C2×SD16×C83C22semidirect
63. #131×C83C22semidirect
64. #108×C83C22semidirect
65. #101×C83C22semidirect
66. #100×C83C22semidirect
95. Dic332D4semidirect
97. #13050580D4semidirect
104. #13050579D4semidirect
105. #13050579D4semidirect
109. #13050578D4semidirect
113. #13050577D4semidirect
116. D664D4semidirect
118. #13050575D4semidirect
228. C83#3291semidirect
237. #471D83semidirect
255. #137C2×D83semidirect
277. C2×SD16C2×D83semidirect
281. #131C2×D83semidirect
285. #108C2×D83semidirect
291. #101C2×D83semidirect
293. #100C2×D83semidirect

Maximal subgroups

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

subgroupindex
#191285132
#191285112
#191285102
#191285072
#191280722
#191279672
#191279652
#191279632
#191278862
#191278822
#191278712
#191278672
#191278172
#191278162
#471×C832
#329183