Finite Group Databasebeta

← #24918249

Subgroups — #24918249

Order 10624.

420 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 115 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#19128214C211yesmaximal, characteristic
2C2×#14581772C211yesmaximal, characteristic, complemented
3#19128045C212yesmaximal, complemented
4#19128014C211yesmaximal, characteristic, complemented
5#19127941C212yesmaximal, complemented
6#19127931C212yesmaximal, complemented
7#19127866C212yesmaximal, complemented
8#19127856C211yesmaximal, characteristic, complemented
9#19127842C211yesmaximal, characteristic, complemented
10C22⋊C8×D83C211yesmaximal, characteristic, complemented
11#483×C83C211yesmaximal, characteristic, complemented
12C2×#13050591C2211yescharacteristic
13C22×Dic166C2211yescharacteristic
14C22×C4×D83C2211yescharacteristic
15C2×#13050580C2212yescomplemented
16C2×#13050579C2212yescomplemented
17C2×C8×D83C2212yes—
18#14581777—21no—
19#14581772—21no—
20#14581772—21no—
21#14581772C2212yes—
22#14581772C2212yes—
23#14581768C2211yescharacteristic
24#14581767C2211yescharacteristic
25#14581766C2211yescharacteristic
26#14581762—21no—
27#14581754—21no—
28(C22⋊D83)⋊C4—21no—
29#14581750C2211yescharacteristic
30C2×C4⋊Dic83—21no—
31C2×(C2×D83)⋊C4—21no—
32#14581725C2212yescomplemented
33#14581717C2212yescomplemented
34#14581715C2212yescomplemented
35#14581712C2212yescomplemented
36C22⋊(C83⋊C8)C2211yescharacteristic
37(C2×D83)⋊C8C2212yes—
38C22⋊Q8×C83C2211yescharacteristic
39C2×C4⋊C4×C83C2211yescharacteristic
40#106×C83C2212yescomplemented
41#103×C83C2212yescomplemented
42C22⋊C8×C83C2211yescharacteristic
43C23×D83D411yescharacteristic
44#13050591—42no—
45C2×Dic166—21no—
46C2×Dic166—42no—
47C2×Dic166C2312yes—
48C2×D332—21no—
49C2×D332—21no—
50C2×D332C2311yescharacteristic
51C2×C22⋊D83—21no—
52C22×Dic83—21no—
53C22×Dic83D411yescharacteristic
54C2×C4×D83—21no—
55C2×C4×D83—42no—
56C2×C4×D83D411yescharacteristic
57C2×C4×D83D411yescharacteristic
58C2×C4×D83C2311yescharacteristic
59C2×C4×D83C2311yescharacteristic
60#13050580—42no—
61#13050579—42no—
62C2×C83⋊C8C2312yes—
63C8×D83—42no—
64C4⋊Dic83—22no—
65C4⋊Dic83—22no—
66C4⋊Dic83C2311yescharacteristic
67Dic83⋊C4—21no—
68(C2×D83)⋊C4—21no—
69(C2×D83)⋊C4—21no—
70(C2×D83)⋊C4—22no—
71(C2×D83)⋊C4—22no—
72(C2×D83)⋊C4—21no—
73(C2×D83)⋊C4—21no—
74C4×Dic83—21no—
75C2×Q8×C83C2311yescharacteristic
76C22×C4×C83—21no—
77C22×C4×C83C2311yescharacteristic
78C2×C8×C83C2312yes—
79C4⋊C4×C83—21no—
80C4⋊C4×C83—21no—
81C4⋊C4×C83C2311yescharacteristic
82C4⋊C4×C83C2312yes—
83C22⋊C4×C83—21no—
84C23×C83C2×D411yescharacteristic
85C22×D83—22no—
86C22×D83—22no—
87C22×D83—21no—
88C22×D83—41no—
89C22×D83C2×D411yescharacteristic
90C22×D83C2×D411yescharacteristic
91Q8×C83—22no—
92C2×C4×C83—22no—
93C2×C4×C83—21no—
94C2×C4×C83—21no—
95C2×C4×C83—22no—
96C2×C4×C83—42no—
97C2×C4×C83C2×D411yescharacteristic
98C2×C4×C83C2411yescharacteristic
99C22⋊D83—42no—
100C2×Dic83—21no—
101C2×Dic83—22no—
102C2×Dic83—41no—
103C2×Dic83—42no—
104C2×Dic83C2×D411yescharacteristic
105C2×Dic83C2×D411yescharacteristic
106D332—22no—
107D332—42no—
108D332—42no—
109C4×D83—22no—
110C4×D83—22no—
111C4×D83—41no—
112C4×D83—41no—
113C4×D83—42no—
114Dic166—22no—
115Dic166—22no—
116Dic166—42no—
117C8×C83—22no—
118C83⋊C8—22no—
119C22×C83—21no—
120C22×C83—21no—
121C22×C83#13512yes—
122C22×C83C22×D411yescharacteristic
123C2×D83—21no—
124C2×D83—21no—
125C2×D83—21no—
126C2×D83—21no—
127C2×D83—21no—
128C2×D83—44no—
129C2×D83—41no—
130C2×D83—41no—
131C2×D83—41no—
132C2×D83#13712yes—
133C4×C83—21no—
134C4×C83—41no—
135C4×C83—42no—
136C4×C83—41no—
137C4×C83H(5,𝔽2)12yes—
138Dic83—21no—
139Dic83—21no—
140Dic83—42no—
141Dic83—41no—
142C2×C83—22no—
143C2×C83#53411yescharacteristic
144C2×C83C2×#13511yescharacteristic
145C2×C83C2×#13711yescharacteristic
146D83—22no—
147D83—41no—
148D83—81no—
149#3190—831nomaximal, Sylow
150C83#319011yescharacteristic, complemented, Sylow
151#525—831no—
152C2×C22⋊Q8—831no—
153#485—832no—
154#483D8311yescharacteristic, complemented
155#480—831no—
156#467—832no—
157#465—832no—
158#452—832no—
159#451—831no—
160#448—831no—
161C2×C22⋊C8—831no—
162C2×#41—831no—
163C22×Q8—831no—
164C23×C4—831no—
165C2×Q16—832no—
166C2×SD16—832no—
167C22×C8—832no—
168#123—1661no—
169C22⋊Q8—831no—
170C22⋊Q8—832no—
171C22⋊Q8—832no—
172C22⋊Q8—1661no—
173C22⋊Q8—1661no—
174C22⋊Q8C2×D8311yescharacteristic
175#121—831no—
176#121—831no—
177#120—1661no—
178C4×D4—1661no—
179C4×D4—1661no—
180#117—831no—
181C2×C4⋊C4—1661no—
182C2×C4⋊C4C2×D8311yescharacteristic
183C2×C22⋊C4—1661no—
184#107—832no—
185#106C2×D8312yescomplemented
186#103—832no—
187#103—832no—
188#103C2×D8312yescomplemented
189#102—832no—
190C22⋊C8—831no—
191C22⋊C8—832no—
192C22⋊C8C2×D8311yescharacteristic
193C24—831no—
194#41—3322no—
195C2×Q8—3322no—
196C2×Q8—832no—
197C2×Q8—1661no—
198C2×Q8C22×D8311yescharacteristic
199C2×D4—831no—
200C2×D4—1661no—
201C2×D4—1661no—
202C2×D4—1661no—
203C22×C4—21no—
204C22×C4—3322no—
205C22×C4—831no—
206C22×C4—831no—
207C22×C4—831no—
208C22×C4—831no—
209C22×C4—831no—
210C22×C4—1661no—
211C22×C4—1661no—
212C22×C4C22×D8311yescharacteristic
213Q16—3322no—
214SD16—3322no—
215C2×C8—3322no—
216C2×C8—832no—
217C2×C8C22×D8312yes—
218C4⋊C4—21no—
219C4⋊C4—21no—
220C4⋊C4—831no—
221C4⋊C4—1662no—
222C4⋊C4—1662no—
223C4⋊C4—1661no—
224C4⋊C4C22×D8312yes—
225C4⋊C4C22×D8311yescharacteristic
226C22⋊C4—21no—
227C22⋊C4—1661no—
228C22⋊C4—1662no—
229C22⋊C4—1662no—
230C22⋊C4—1661no—
231C22⋊C4—1661no—
232C22⋊C4—1661no—
233C42—1661no—
234C23—3321no—
235C23—831no—
236C23—831no—
237C23—1662no—
238C23—1661no—
239C23—1662no—
240C23D4×D8311yescharacteristic
241Q8—22no—
242Q8—3322no—
243Q8—1662no—
244Q8—1662no—
245D4—3322no—
246D4—3322no—
247D4—3322no—
248D4—1662no—
249C2×C4—22no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—22no—
253C2×C4—42no—
254C2×C4—3321no—
255C2×C4—3321no—
256C2×C4—3322no—
257C2×C4—3322no—
258C2×C4—3321no—
259C2×C4—831no—
260C2×C4—831no—
261C2×C4—1661no—
262C2×C4—1662no—
263C2×C4—1662no—
264C2×C4—1662no—
265C2×C4D4×D8311yescharacteristic
266C2×C4C23×D8311yescharacteristic
267C8—22no—
268C8—1662no—
269C22—21no—
270C22—21no—
271C22—3324no—
272C22—3321no—
273C22—3321no—
274C22—3321no—
275C22—832no—
276C22—1661no—
277C22—1661no—
278C22—1661no—
279C22—1661no—
280C22—1661no—
281C22#1458181312yes—
282C22C2×D4×D8311yescharacteristic
283C4—21no—
284C4—42no—
285C4—41no—
286C4—41no—
287C4—3321no—
288C4—3322no—
289C4—1661no—
290C4—1661no—
291C4#1458183812yes—
292C2—22no—
293C2—3321no—
294C2—6641no—
295C2—1662no—
296C2#1912826211yescharacteristic
297C2C2×#1458181311yescharacteristic
298C2#137×D8311yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#483×C83C2
socleC22×C83C22×D4
prime radical#19128214C2
hypercenter#483D83
hypocenterC83#3190
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ #483

Complements

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

NH ≅ G/Ntype
2. C2×#14581772C2semidirect
3. #19128045C2semidirect
4. #19128014C2semidirect
5. #19127941C2semidirect
6. #19127931C2semidirect
7. #19127866C2semidirect
8. #19127856C2semidirect
9. #19127842C2semidirect
10. C22⋊C8×D83C2semidirect
11. #483×C83C2semidirect
15. C2×#13050580C22semidirect
16. C2×#13050579C22semidirect
32. #14581725C22semidirect
33. #14581717C22semidirect
34. #14581715C22semidirect
35. #14581712C22semidirect
40. #106×C83C22semidirect
41. #103×C83C22semidirect
150. C83#3190semidirect
154. #483D83semidirect
185. #106C2×D83semidirect
188. #103C2×D83semidirect

Maximal subgroups

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

subgroupindex
#191282142
C2×#145817722
#191280452
#191280142
#191279412
#191279312
#191278662
#191278562
#191278422
C22⋊C8×D832
#483×C832
#319083