Finite Group Databasebeta

← #24919261

Subgroups — #24919261

Order 10624.

476 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 117 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#19128506C211yesmaximal, characteristic, complemented
2#19128505C212yesmaximal, complemented
3#19128502C211yesmaximal, characteristic, complemented
4#19128076C211yesmaximal, characteristic, complemented
5#19127972C211yesmaximal, characteristic, complemented
6#19127970C212yesmaximal, complemented
7#19127871C212yesmaximal, complemented
8#19127867C212yesmaximal, complemented
9#19127817C211yesmaximal, characteristic, complemented
10#19127816C211yesmaximal, characteristic, complemented
11#472×C83C211yesmaximal, characteristic, complemented
12#14581838C2212yescomplemented
13#14581837C2212yescomplemented
14#14581820—22no—
15#14581820C2212yescomplemented
16#14581820C2212yescomplemented
17#14581819—22no—
18#14581818C2212yescomplemented
19#14581816—22no—
20#14581816—22no—
21#14581814—21no—
22#14581814C2211yescharacteristic
23#14581813—22no—
24#14581812C2211yescharacteristic
25#14581811—21no—
26C2×D664—21no—
27C2×D664C2211yescharacteristic, complemented
28C2×#13050575C2212yescomplemented
29D8×D83—21no—
30D8×D83—22no—
31#14581800C2212yescomplemented
32#14581798C2211yescharacteristic, complemented
33#14581788C2212yescomplemented
34#14581727C2211yescharacteristic, complemented
35#14581719C2211yescharacteristic, complemented
36#14581718C2211yescharacteristic, complemented
37#14581718C2212yescomplemented
38#14581708C2212yescomplemented
39C8×Dic83C2211yescharacteristic, complemented
40AGL(1,ℤ/8)×C83C2212yescomplemented
41C2×D8×C83C2211yescharacteristic, complemented
42#131×C83C2211yescharacteristic, complemented
43#108×C83C2211yescharacteristic, complemented
44#100×C83C2212yescomplemented
45#13050591—22no—
46#13050590—22no—
47#13050590C2311yescharacteristic
48#13050590C2312yes—
49#13050589—22no—
50#13050589—22no—
51C2×D332—22no—
52C2×D332C2312yes—
53C2×C22⋊D83—42no—
54C2×C22⋊D83—42no—
55D4×D83—22no—
56D4×D83—22no—
57D4×D83—22no—
58D4×D83—42no—
59Dic332D411yescharacteristic, complemented
60#13050580D411yescharacteristic, complemented
61#13050579—22no—
62#13050579—22no—
63#13050579D412yescomplemented
64#13050578—22no—
65#13050578D411yescharacteristic, complemented
66#13050577D411yescharacteristic, complemented
67D664—22no—
68D664—22no—
69D664—21no—
70D664—21no—
71D664D411yescharacteristic, complemented
72#13050575—22no—
73#13050575—22no—
74#13050575—22no—
75#13050575D411yescharacteristic, complemented
76#13050574C2311yescharacteristic
77#13050573—21no—
78#13050573—22no—
79C2×C83⋊C8C2311yescharacteristic
80C8×D83—21no—
81C8×D83—22no—
82C4×Dic83C2311yescharacteristic
83#41×C83C2311yescharacteristic
84C2×D4×C83C2312yes—
85SD16×C83—22no—
86D8×C83—22no—
87D8×C83—21no—
88D8×C83—21no—
89M16×C83—21no—
90M16×C83C2312yes—
91M16×C83C2311yescharacteristic
92C2×C8×C83—21no—
93C2×C8×C83C2311yescharacteristic
94C23×C83—22no—
95C22×D83—22no—
96C22×D83—42no—
97C22×D83—42no—
98Q8×C83C2×D411yescharacteristic
99D4×C83—21no—
100D4×C83—22no—
101D4×C83—22no—
102D4×C83C2×D411yescharacteristic
103C2×C4×C83—21no—
104C2×C4×C83C2411yescharacteristic
105C22⋊D83—22no—
106C22⋊D83—21no—
107C22⋊D83—42no—
108C22⋊D83—42no—
109C22⋊D83—42no—
110C22⋊D83—84no—
111C2×Dic83—21no—
112C2×Dic83—42no—
113D332—22no—
114D332—22no—
115D332—42no—
116D332C2×D412yes—
117D332C2×D411yescharacteristic
118C4×D83—21no—
119C4×D83—22no—
120C4×D83—42no—
121Dic166C2×D412yes—
122Dic166C2×D411yescharacteristic
123C8×C83—21no—
124C8×C83—22no—
125C8×C83C2×D411yescharacteristic
126C8×C83C2×D411yescharacteristic
127C83⋊C8—21no—
128C83⋊C8C2×D411yescharacteristic
129C83⋊C8C2×D411yescharacteristic
130C22×C83—21no—
131C22×C83—42no—
132C22×C83—82no—
133C22×C83H(5,𝔽2)11yescharacteristic
134C2×D83—22no—
135C2×D83—21no—
136C2×D83—42no—
137C2×D83—82no—
138C2×D83—162no—
139C2×D83—162no—
140C4×C83—21no—
141C4×C83C22×D411yescharacteristic
142C4×C83C22×D411yescharacteristic
143Dic83—22no—
144Dic83—22no—
145Dic83—21no—
146C2×C83—21no—
147C2×C83—41no—
148C2×C83—82no—
149C2×C83D4211yescharacteristic
150D83—42no—
151D83—41no—
152D83—82no—
153#3292—831nomaximal, Sylow
154C83#329211yescharacteristic, complemented, Sylow
155#576—832no—
156#576—831no—
157#576—831no—
158#493—831no—
159#472—831no—
160#472—832no—
161#472D8311yescharacteristic, complemented
162#453—832no—
163#453—832no—
164#443—831no—
165#443—831no—
166H(5,𝔽2)—832no—
167H(5,𝔽2)—832no—
168AGL(1,ℤ/8)—832no—
169AGL(1,ℤ/8)—832no—
170AGL(1,ℤ/8)—832no—
171AGL(1,ℤ/8)—1662no—
172AGL(1,ℤ/8)—1662no—
173AGL(1,ℤ/8)—1662no—
174AGL(1,ℤ/8)—1662no—
175AGL(1,ℤ/8)C2×D8312yescomplemented
176#135—831no—
177#135—831no—
178#135—1661no—
179#135—1661no—
180#135—1662no—
181C2×D8—832no—
182C2×D8—831no—
183C2×D8—1661no—
184C2×D8—1662no—
185C2×D8—1661no—
186C2×D8C2×D8311yescharacteristic, complemented
187#131—832no—
188#131—831no—
189#131C2×D8311yescharacteristic, complemented
190#127—832no—
191#108—831no—
192#108C2×D8311yescharacteristic, complemented
193#104—831no—
194#104—832no—
195#104—831no—
196#100—832no—
197#100C2×D8312yescomplemented
198C4×C8—831no—
199#41—832no—
200#41—831no—
201#41—1662no—
202#41—1662no—
203#41—1662no—
204#41—1662no—
205#41C22×D8311yescharacteristic
206C2×D4—3322no—
207C2×D4—3322no—
208C2×D4—3322no—
209C2×D4—832no—
210C2×D4—1662no—
211C2×D4—1662no—
212C2×D4—1662no—
213C2×D4—1662no—
214C2×D4C22×D8312yes—
215Q16—831no—
216Q16—831no—
217SD16—22no—
218SD16—831no—
219SD16—831no—
220SD16—832no—
221SD16—1662no—
222SD16—1662no—
223SD16—1662no—
224D8—22no—
225D8—21no—
226D8—21no—
227D8—831no—
228D8—831no—
229D8—1662no—
230D8—1662no—
231D8—1662no—
232D8—1662no—
233D8—1662no—
234D8—1661no—
235D8—1661no—
236M16—21no—
237M16—831no—
238M16—1662no—
239M16—1661no—
240M16C22×D8312yes—
241M16C22×D8311yescharacteristic
242C2×C8—21no—
243C2×C8—831no—
244C2×C8—1662no—
245C2×C8—1661no—
246C2×C8C22×D8311yescharacteristic
247C42—831no—
248C23—22no—
249C23—3322no—
250C23—3322no—
251C23—1662no—
252Q8—831no—
253Q8—832no—
254Q8D4×D8311yescharacteristic
255D4—21no—
256D4—22no—
257D4—22no—
258D4—3322no—
259D4—3322no—
260D4—3322no—
261D4—3322no—
262D4—831no—
263D4—832no—
264D4—6644no—
265D4—1662no—
266D4—1661no—
267D4—1662no—
268D4—1662no—
269D4D4×D8311yescharacteristic
270C2×C4—21no—
271C2×C4—3322no—
272C2×C4—3322no—
273C2×C4—1661no—
274C2×C4—1662no—
275C2×C4—1661no—
276C2×C4C23×D8311yescharacteristic
277C8—21no—
278C8—22no—
279C8—831no—
280C8—831no—
281C8—1661no—
282C8D4×D8311yescharacteristic
283C8D4×D8311yescharacteristic
284C22—21no—
285C22—42no—
286C22—82no—
287C22—13282no—
288C22—13282no—
289C22—3322no—
290C22—6642no—
291C22—1662no—
292C22—1661no—
293C22#1458183811yescharacteristic
294C4—21no—
295C4—1662no—
296C4—1662no—
297C4—1661no—
298C4C2×D4×D8311yescharacteristic
299C4C2×D4×D8311yescharacteristic
300C2—21no—
301C2—41no—
302C2—82no—
303C2—3322no—
304C2—3321no—
305C2—6642no—
306C2#1912833511yescharacteristic
subgroupG/H
centerC2#19128335
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#472×C83C2
socleC2×C83D42
hypercenter#472D83
hypocenterC83#3292
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C2 ◁ C2×C4 ◁ #472

Complements

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

NH ≅ G/Ntype
1. #19128506C2semidirect
2. #19128505C2semidirect
3. #19128502C2semidirect
4. #19128076C2semidirect
5. #19127972C2semidirect
6. #19127970C2semidirect
7. #19127871C2semidirect
8. #19127867C2semidirect
9. #19127817C2semidirect
10. #19127816C2semidirect
11. #472×C83C2semidirect
12. #14581838C22semidirect
13. #14581837C22semidirect
15. #14581820C22semidirect
16. #14581820C22semidirect
18. #14581818C22semidirect
27. C2×D664C22semidirect
28. C2×#13050575C22semidirect
31. #14581800C22semidirect
32. #14581798C22semidirect
33. #14581788C22semidirect
34. #14581727C22semidirect
35. #14581719C22semidirect
36. #14581718C22semidirect
37. #14581718C22semidirect
38. #14581708C22semidirect
39. C8×Dic83C22semidirect
40. AGL(1,ℤ/8)×C83C22semidirect
41. C2×D8×C83C22semidirect
42. #131×C83C22semidirect
43. #108×C83C22semidirect
44. #100×C83C22semidirect
59. Dic332D4semidirect
60. #13050580D4semidirect
63. #13050579D4semidirect
65. #13050578D4semidirect
66. #13050577D4semidirect
71. D664D4semidirect
75. #13050575D4semidirect
154. C83#3292semidirect
161. #472D83semidirect
175. AGL(1,ℤ/8)C2×D83semidirect
186. C2×D8C2×D83semidirect
189. #131C2×D83semidirect
192. #108C2×D83semidirect
197. #100C2×D83semidirect

Maximal subgroups

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

subgroupindex
#191285062
#191285052
#191285022
#191280762
#191279722
#191279702
#191278712
#191278672
#191278172
#191278162
#472×C832
#329283