Finite Group Databasebeta

← #24919251

Subgroups — #24919251

Order 10624.

454 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 125 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#19128515C211yesmaximal, characteristic, complemented
2#19128513C212yesmaximal, complemented
3#19128503C211yesmaximal, characteristic, complemented
4#19128085C211yesmaximal, characteristic, complemented
5#19127977C211yesmaximal, characteristic, complemented
6#19127972C211yesmaximal, characteristic, complemented
7#19127964C212yesmaximal, complemented
8#19127880C212yesmaximal, complemented
9#19127875C212yesmaximal, complemented
10#19127819C211yesmaximal, characteristic, complemented
11#443×C83C211yesmaximal, characteristic, complemented
12#14581839C2212yescomplemented
13#14581838C2212yes—
14#14581825—22no—
15#14581825C2212yes—
16#14581822C2212yescomplemented
17#14581821—22no—
18#14581820—22no—
19#14581820C2212yescomplemented
20#14581819—22no—
21#14581817C2212yes—
22#14581815—21no—
23#14581814—22no—
24#14581814C2211yescharacteristic
25#14581811—21no—
26C2×Dic332C2211yescharacteristic, complemented
27Q16×D83—21no—
28C2×#13050579—22no—
29C2×#13050579C2212yescomplemented
30C2×D664C2211yescharacteristic
31D8×D83—21no—
32#14581798C2211yescharacteristic
33#14581798C2211yescharacteristic, complemented
34#14581780C2212yescomplemented
35#14581728C2211yescharacteristic, complemented
36#14581720C2211yescharacteristic
37#14581720C2211yescharacteristic, complemented
38#14581710C2212yescomplemented
39#14581708C2212yescomplemented
40#135×C83C2211yescharacteristic
41#131×C83C2212yescomplemented
42#108×C83C2211yescharacteristic, complemented
43#104×C83C2212yescomplemented
44C4×C8×C83C2211yescharacteristic, complemented
45#13050591—22no—
46#13050590—22no—
47#13050590—22no—
48#13050590C2311yescharacteristic
49#13050590C2311yescharacteristic
50#13050589—22no—
51#13050589—42no—
52C2×Dic166—22no—
53C2×Dic166C2312yes—
54Q8×D83—22no—
55C2×D332—22no—
56C2×D332C2312yes—
57D4×D83—22no—
58D4×D83—42no—
59Dic332—22no—
60Dic332—21no—
61Dic332—21no—
62Dic332D411yescharacteristic
63#13050580—22no—
64#13050579—22no—
65#13050579—22no—
66#13050579—22no—
67#13050579—22no—
68#13050579D411yescharacteristic
69#13050579D411yescharacteristic
70#13050578—22no—
71#13050577—22no—
72D664—22no—
73D664—21no—
74D664—21no—
75D664D411yescharacteristic
76#13050575—22no—
77#13050574C2311yescharacteristic
78#13050574C2311yescharacteristic
79#13050573—21no—
80#13050573—21no—
81C8×D83—21no—
82C8×D83—21no—
83(C2×D83)⋊C4—42no—
84#41×C83C2312yes—
85Q16×C83D411yescharacteristic
86SD16×C83D412yes—
87D8×C83D411yescharacteristic
88M16×C83—22no—
89M16×C83C2312yes—
90C2×C8×C83—22no—
91C2×C8×C83C2311yescharacteristic
92C2×C8×C83C2311yescharacteristic
93C42×C83C2311yescharacteristic
94C22×D83—22no—
95C22×D83—42no—
96Q8×C83C2×D412yes—
97D4×C83—22no—
98D4×C83C2×D412yes—
99C2×C4×C83—22no—
100C2×C4×C83—21no—
101C2×C4×C83C2411yescharacteristic
102C22⋊D83—21no—
103C22⋊D83—21no—
104C22⋊D83—42no—
105C22⋊D83—42no—
106C2×Dic83—22no—
107C2×Dic83—42no—
108D332—22no—
109D332—22no—
110D332—42no—
111D332C2×D411yescharacteristic
112D332C2×D411yescharacteristic
113C4×D83—21no—
114C4×D83—21no—
115C4×D83—42no—
116C4×D83—42no—
117Dic166—22no—
118Dic166—22no—
119Dic166—42no—
120Dic166C2×D411yescharacteristic
121Dic166C2×D411yescharacteristic
122C8×C83—22no—
123C8×C83C2×D411yescharacteristic
124C8×C83C2×D411yescharacteristic
125C8×C83C2×D411yescharacteristic
126C8×C83C2×D411yescharacteristic
127C83⋊C8—21no—
128C83⋊C8—21no—
129C22×C83—22no—
130C22×C83H(5,𝔽2)11yescharacteristic
131C2×D83—21no—
132C2×D83—21no—
133C2×D83—42no—
134C2×D83—82no—
135C2×D83—162no—
136C4×C83—22no—
137C4×C83—41no—
138C4×C83C22×D411yescharacteristic
139C4×C83C22×D411yescharacteristic
140Dic83—21no—
141Dic83—21no—
142Dic83—42no—
143C2×C83—21no—
144C2×C83—42no—
145C2×C83D4211yescharacteristic
146D83—41no—
147D83—41no—
148D83—82no—
149#3290—831nomaximal, Sylow
150C83#329011yescharacteristic, complemented, Sylow
151#578—831no—
152#577—832no—
153#576—831no—
154#495—831no—
155#473—831no—
156#472—831no—
157#471—832no—
158#455—832no—
159#454—832no—
160#443—831no—
161#443D8311yescharacteristic, complemented
1622−1+4—832no—
163H(5,𝔽2)—832no—
164#137—832no—
165#137—832no—
166#137—1662no—
167#137—1662no—
168AGL(1,ℤ/8)—832no—
169AGL(1,ℤ/8)—832no—
170AGL(1,ℤ/8)—1662no—
171AGL(1,ℤ/8)—1662no—
172#135—831no—
173#135—1662no—
174#135—1661no—
175#135—1661no—
176#135C2×D8311yescharacteristic
177C2×Q16—831no—
178C2×Q16—1661no—
179C2×SD16—832no—
180C2×SD16—1662no—
181C2×D8—831no—
182C2×D8—1661no—
183#131—831no—
184#131—831no—
185#131C2×D8312yescomplemented
186#124—832no—
187#108—831no—
188#108C2×D8311yescharacteristic, complemented
189#104—831no—
190#104—831no—
191#104C2×D8312yescomplemented
192#101—832no—
193#100—832no—
194C4×C8C2×D8311yescharacteristic, complemented
195#41—3322no—
196#41—831no—
197#41—831no—
198#41—1662no—
199#41—1662no—
200#41—1662no—
201#41—1662no—
202#41C22×D8312yes—
203C2×Q8—832no—
204C2×Q8—1662no—
205C2×Q8—1662no—
206C2×D4—3322no—
207C2×D4—832no—
208C2×D4—1662no—
209C2×D4—1662no—
210Q16—831no—
211Q16—1662no—
212Q16—1662no—
213Q16—1661no—
214Q16—1661no—
215Q16D33211yescharacteristic
216SD16—831no—
217SD16—831no—
218SD16—1662no—
219SD16—1662no—
220SD16—1662no—
221SD16—1662no—
222SD16—1662no—
223SD16—1662no—
224SD16D33212yes—
225D8—831no—
226D8—1662no—
227D8—1662no—
228D8—1661no—
229D8—1661no—
230D8D33211yescharacteristic
231M16—22no—
232M16—831no—
233M16—831no—
234M16—1661no—
235M16—1661no—
236M16C22×D8312yes—
237C2×C8—22no—
238C2×C8—1661no—
239C2×C8—1661no—
240C2×C8C22×D8311yescharacteristic
241C2×C8C22×D8311yescharacteristic
242C22⋊C4—3322no—
243C42C22×D8311yescharacteristic
244C23—3322no—
245C23—1662no—
246Q8—3322no—
247Q8—831no—
248Q8—831no—
249Q8—1662no—
250Q8—1662no—
251Q8C2×D33212yes—
252D4—22no—
253D4—3322no—
254D4—3322no—
255D4—3322no—
256D4—831no—
257D4—831no—
258D4—1661no—
259D4—1661no—
260D4—1662no—
261D4—1662no—
262D4C2×D33212yes—
263C2×C4—22no—
264C2×C4—21no—
265C2×C4—3322no—
266C2×C4—3322no—
267C2×C4—3322no—
268C2×C4—1662no—
269C2×C4—1661no—
270C2×C4—1661no—
271C2×C4C23×D8311yescharacteristic
272C8—22no—
273C8—1661no—
274C8—1661no—
275C8D4×D8311yescharacteristic
276C8D4×D8311yescharacteristic
277C8C2×D33211yescharacteristic
278C8C2×D33211yescharacteristic
279C22—22no—
280C22—13282no—
281C22—3322no—
282C22—6642no—
283C22—1661no—
284C22—1661no—
285C22#1458183811yescharacteristic
286C4—22no—
287C4—41no—
288C4—3322no—
289C4—1661no—
290C4—1661no—
291C4C2×D4×D8311yescharacteristic
292C4C22×D33211yescharacteristic
293C2—21no—
294C2—42no—
295C2—3321no—
296C2—3321no—
297C2—6642no—
298C2D4×D33211yescharacteristic
subgroupG/H
centerC2D4×D332
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#443×C83C2
socleC2×C83D42
hypercenter#443D83
hypocenterC83#3290
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C2 ◁ C2×C4 ◁ #443

Complements

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

NH ≅ G/Ntype
1. #19128515C2semidirect
2. #19128513C2semidirect
3. #19128503C2semidirect
4. #19128085C2semidirect
5. #19127977C2semidirect
6. #19127972C2semidirect
7. #19127964C2semidirect
8. #19127880C2semidirect
9. #19127875C2semidirect
10. #19127819C2semidirect
11. #443×C83C2semidirect
12. #14581839C22semidirect
16. #14581822C22semidirect
19. #14581820C22semidirect
26. C2×Dic332C22semidirect
29. C2×#13050579C22semidirect
33. #14581798C22semidirect
34. #14581780C22semidirect
35. #14581728C22semidirect
37. #14581720C22semidirect
38. #14581710C22semidirect
39. #14581708C22semidirect
41. #131×C83C22semidirect
42. #108×C83C22semidirect
43. #104×C83C22semidirect
44. C4×C8×C83C22semidirect
150. C83#3290semidirect
161. #443D83semidirect
185. #131C2×D83semidirect
188. #108C2×D83semidirect
191. #104C2×D83semidirect
194. C4×C8C2×D83semidirect

Maximal subgroups

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

subgroupindex
#191285152
#191285132
#191285032
#191280852
#191279772
#191279722
#191279642
#191278802
#191278752
#191278192
#443×C832
#329083