Finite Group Databasebeta

← #24919082

Subgroups — #24919082

Order 10624.

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

normal subgroups only

This group has 556 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
1C2×#14581816C212yesmaximal, complemented
2#19128371C211yesmaximal, characteristic, complemented
3#19128336C211yesmaximal, characteristic, complemented
4#19127951C212yesmaximal, complemented
5#19127898C211yesmaximal, characteristic, complemented
6#19127893C211yesmaximal, characteristic, complemented
7#19127843C212yesmaximal, complemented
8#19127833C212yesmaximal, complemented
9C4×#13050579C211yesmaximal, characteristic, complemented
10#19127780C211yesmaximal, characteristic, complemented
11#493×C83C211yesmaximal, characteristic, complemented
12C2×#13050589C2212yescomplemented
13C2×D4×D83—21no—
14C2×D4×D83C2212yes—
15#14581816—28no—
16C2×#13050579C2211yescharacteristic
17C2×#13050578C2212yescomplemented
18C2×#13050575C2212yes—
19C2×#13050573C2212yescomplemented
20#14581788—21no—
21#14581764—22no—
22#14581763—21no—
23#14581763C2212yes—
24#14581763C2212yescomplemented
25#14581760—22no—
26C4×Dic166C2211yescharacteristic, complemented
27C4×D332C2211yescharacteristic
28C4×D332C2211yescharacteristic
29D4×Dic83—21no—
30#14581724C2211yescharacteristic, complemented
31C4⋊(C83⋊C8)C2211yescharacteristic, complemented
32#14581717C2211yescharacteristic, complemented
33#14581714C2211yescharacteristic
34#14581713C2212yescomplemented
35#14581713C2212yescomplemented
36(C2×D83)⋊C8C2212yescomplemented
37C2×D8×C83C2212yescomplemented
38C2×D8×C83C2212yescomplemented
39#127×C83C2211yescharacteristic
40#127×C83C2211yescharacteristic, complemented
41C4×C8×C83C2211yescharacteristic, complemented
42C23×D83—22no—
43#13050589—24no—
44#13050589—42no—
45C2×Dic166C2311yescharacteristic
46C2×D332D411yescharacteristic
47C2×D332C2311yescharacteristic
48C2×C22⋊D83—22no—
49C2×C22⋊D83—22no—
50C2×C22⋊D83—22no—
51C2×C22⋊D83—22no—
52D4×D83—24no—
53D4×D83—42no—
54D4×D83—44no—
55C22×Dic83—22no—
56C2×C4×D83—21no—
57C2×C4×D83C2312yes—
58#13050579D414yescomplemented
59#13050578—24no—
60#13050575—24no—
61#13050573—24no—
62C2×C83⋊C8C2312yes—
63C4⋊Dic83D411yescharacteristic
64C4⋊Dic83C2311yescharacteristic
65C4⋊Dic83C2311yescharacteristic
66Dic83⋊C4—21no—
67C22⋊Dic83—22no—
68C22⋊Dic83—22no—
69(C2×D83)⋊C4—21no—
70(C2×D83)⋊C4D412yes—
71C4×Dic83—21no—
72C2×D4×C83—21no—
73C2×D4×C83—21no—
74C2×D4×C83—21no—
75C2×D4×C83—21no—
76C2×D4×C83C2312yes—
77C2×D4×C83C2312yes—
78D8×C83—24no—
79D8×C83—42no—
80C2×C8×C83C2311yescharacteristic
81C2×C8×C83C2311yescharacteristic
82C42×C83C2311yescharacteristic
83C23×C83—22no—
84C23×C83—22no—
85C22×D83—21no—
86C22×D83—44no—
87C22×D83—44no—
88C22×D83—82no—
89C22×D83C2×D412yes—
90D4×C83—22no—
91D4×C83—22no—
92D4×C83—22no—
93D4×C83—22no—
94D4×C83—42no—
95D4×C83—42no—
96D4×C83—81no—
97D4×C83—81no—
98C2×C4×C83C2×D411yescharacteristic
99C2×C4×C83C2×D411yescharacteristic
100C2×C4×C83C2411yescharacteristic
101C22⋊D83—44no—
102C22⋊D83—44no—
103C22⋊D83—44no—
104C22⋊D83—82no—
105C2×Dic83—21no—
106C2×Dic83—21no—
107C2×Dic83—21no—
108C2×Dic83—44no—
109C2×Dic83C2×D412yes—
110D332—22no—
111D332—21no—
112D332C2×D412yes—
113C4×D83—22no—
114C4×D83—22no—
115C4×D83—42no—
116Dic166—21no—
117Dic166C2×D412yescomplemented
118C8×C83—21no—
119C8×C83C2×D412yescomplemented
120C83⋊C8—22no—
121C22×C83—42no—
122C22×C83—42no—
123C22×C83—42no—
124C22×C83—42no—
125C22×C83—42no—
126C22×C83—42no—
127C22×C83C22×D411yescharacteristic
128C2×D83—22no—
129C2×D83—22no—
130C2×D83—22no—
131C2×D83—22no—
132C2×D83—41no—
133C2×D83—41no—
134C2×D83—88no—
135C4×C83—21no—
136C4×C83AGL(1,ℤ/8)12yescomplemented
137C4×C83C22×D411yescharacteristic
138C4×C83H(5,𝔽2)11yescharacteristic
139Dic83—22no—
140Dic83—22no—
141Dic83—41no—
142Dic83—41no—
143C2×C83—82no—
144C2×C83—82no—
145C2×C83D4211yescharacteristic
146C2×C83C2×AGL(1,ℤ/8)11yescharacteristic
147C2×C83#57611yescharacteristic
148D83—42no—
149D83—42no—
150#3274—831nomaximal, Sylow
151C83#327411yescharacteristic, complemented, Sylow
152C2×AGL(1,ℤ/8)—832no—
153#550—831no—
154D42—831no—
155#493D8311yescharacteristic, complemented
156#469—832no—
157#460—831no—
158#459—831no—
159#449—832no—
160#447—832no—
161C4×SD16—831no—
162#436—831no—
163C2×#41—832no—
164C22×D4—832no—
165C22×D4—1661no—
166AGL(1,ℤ/8)—1668no—
167C2×SD16—832no—
168C2×SD16—831no—
169C2×D8—832no—
170C2×D8C2×D8312yescomplemented
171C2×D8C2×D8312yescomplemented
172C2×M16—832no—
173#127—1661no—
174#127C2×D8311yescharacteristic
175#127C2×D8311yescharacteristic, complemented
176#121—832no—
177#121—832no—
178#121—1662no—
179#121—1661no—
180#120—1662no—
181C4×Q8—831no—
182C4×D4—831no—
183C4×D4—831no—
184C4×D4—1661no—
185#106—831no—
186C4⋊C8—831no—
187#103—831no—
188#102—832no—
189#102—832no—
190#102—831no—
191C22⋊C8—832no—
192C4×C8C2×D8311yescharacteristic, complemented
193C24—1662no—
194#41—3322no—
195#41—1664no—
196C2×Q8—831no—
197C2×D4—21no—
198C2×D4—21no—
199C2×D4—21no—
200C2×D4—21no—
201C2×D4—3324no—
202C2×D4—3322no—
203C2×D4—831no—
204C2×D4—831no—
205C2×D4—1662no—
206C2×D4—1664no—
207C2×D4—1662no—
208C2×D4—1662no—
209C2×D4—1662no—
210C2×D4C22×D8312yes—
211C2×D4C22×D8312yes—
212C22×C4—832no—
213C22×C4—1662no—
214C22×C4—1661no—
215SD16—834no—
216SD16—1664no—
217D8—24no—
218D8—42no—
219D8—1664no—
220M16—1664no—
221C2×C8—832no—
222C2×C8C22×D8311yescharacteristic
223C2×C8C22×D8311yescharacteristic
224C4⋊C4—831no—
225C4⋊C4—831no—
226C4⋊C4—831no—
227C4⋊C4—1661no—
228C22⋊C4—832no—
229C22⋊C4—1662no—
230C22⋊C4—1662no—
231C22⋊C4—1661no—
232C42—1661no—
233C42C22×D8311yescharacteristic
234C23—22no—
235C23—22no—
236C23—3324no—
237C23—3324no—
238C23—832no—
239C23—6642no—
240C23—1661no—
241Q8—832no—
242Q8—1661no—
243D4—22no—
244D4—22no—
245D4—22no—
246D4—22no—
247D4—42no—
248D4—42no—
249D4—81no—
250D4—81no—
251D4—3324no—
252D4—3324no—
253D4—3324no—
254D4—832no—
255D4—6642no—
256D4—1662no—
257D4—1661no—
258C2×C4—3322no—
259C2×C4—3324no—
260C2×C4—832no—
261C2×C4—1661no—
262C2×C4—1661no—
263C2×C4—1661no—
264C2×C4—1662no—
265C2×C4—1662no—
266C2×C4D4×D8311yescharacteristic
267C2×C4D4×D8311yescharacteristic
268C2×C4C23×D8311yescharacteristic
269C8—21no—
270C8—1662no—
271C8D4×D8312yescomplemented
272C22—42no—
273C22—42no—
274C22—42no—
275C22—42no—
276C22—42no—
277C22—42no—
278C22—3321no—
279C22—3321no—
280C22—6648no—
281C22—1662no—
282C22—1662no—
283C22—1662no—
284C22—1662no—
285C22C2×D4×D8311yescharacteristic
286C4—21no—
287C4—3321no—
288C4—3321no—
289C4—1662no—
290C4—1662no—
291C4#1458181612yescomplemented
292C4C2×D4×D8311yescharacteristic
293C4#1458183711yescharacteristic
294C2—82no—
295C2—82no—
296C2—3322no—
297C2—3322no—
298C2#1912833611yescharacteristic
299C2C2×#1458181611yescharacteristic
300C2#1912850211yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#493×C83C2
socleC22×C83C22×D4
hypercenter#493D83
hypocenterC83#3274
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C42 ◁ #493

Complements

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

NH ≅ G/Ntype
1. C2×#14581816C2semidirect
2. #19128371C2semidirect
3. #19128336C2semidirect
4. #19127951C2semidirect
5. #19127898C2semidirect
6. #19127893C2semidirect
7. #19127843C2semidirect
8. #19127833C2semidirect
9. C4×#13050579C2semidirect
10. #19127780C2semidirect
11. #493×C83C2semidirect
12. C2×#13050589C22semidirect
17. C2×#13050578C22semidirect
19. C2×#13050573C22semidirect
24. #14581763C22semidirect
26. C4×Dic166C22semidirect
30. #14581724C22semidirect
31. C4⋊(C83⋊C8)C22semidirect
32. #14581717C22semidirect
34. #14581713C22semidirect
35. #14581713C22semidirect
36. (C2×D83)⋊C8C22semidirect
37. C2×D8×C83C22semidirect
38. C2×D8×C83C22semidirect
40. #127×C83C22semidirect
41. C4×C8×C83C22semidirect
58. #13050579D4semidirect
117. Dic166C2×D4semidirect
119. C8×C83C2×D4semidirect
136. C4×C83AGL(1,ℤ/8)semidirect
151. C83#3274semidirect
155. #493D83semidirect
170. C2×D8C2×D83semidirect
171. C2×D8C2×D83semidirect
175. #127C2×D83semidirect
192. C4×C8C2×D83semidirect
271. C8D4×D83semidirect
291. C4#14581816semidirect

Maximal subgroups

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

subgroupindex
C2×#145818162
#191283712
#191283362
#191279512
#191278982
#191278932
#191278432
#191278332
C4×#130505792
#191277802
#493×C832
#327483