Finite Group Databasebeta

← #24918246

Subgroups — #24918246

Order 10624.

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

normal subgroups only

This group has 420 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#19128212C211yesmaximal, characteristic
2C2×#14581771C211yesmaximal, characteristic, complemented
3#19128043C212yesmaximal, complemented
4#19128032C211yesmaximal, characteristic, complemented
5#19128015C211yesmaximal, characteristic, complemented
6#19127942C212yesmaximal, complemented
7#19127930C212yesmaximal, complemented
8#19127865C212yesmaximal, complemented
9#19127859C211yesmaximal, characteristic, complemented
10#19127842C211yesmaximal, characteristic, complemented
11C2×C22⋊C8×C83C211yesmaximal, characteristic, complemented
12C2×#13050590C2211yescharacteristic
13C22×Dic166C2211yescharacteristic
14C2×Dic332C2212yescomplemented
15C2×#13050579C2212yescomplemented
16#14581775—21no—
17#14581771—21no—
18#14581771—21no—
19#14581771C2211yescharacteristic
20#14581771C2212yes—
21#14581771C2212yes—
22#14581771C2211yescharacteristic
23#14581766C2211yescharacteristic
24#14581766C2211yescharacteristic
25#14581761—21no—
26C4×C22⋊D83—21no—
27C4×C22⋊D83—21no—
28#14581751C2211yescharacteristic
29C2×C4⋊Dic83C2211yescharacteristic
30C2×Dic83⋊C4—21no—
31C2×C22⋊Dic83—21no—
32#14581726C2212yescomplemented
33#14581724C2212yescomplemented
34#14581717C2212yescomplemented
35#14581717C2212yescomplemented
36#14581717C2212yescomplemented
37#14581714C2212yescomplemented
38C23×C4×C83C2211yescharacteristic
39C22×C8×C83C2212yes—
40C22⋊C8×C83C2211yescharacteristic
41C22⋊C8×C83C2211yescharacteristic
42C22⋊C8×C83C2212yes—
43#13050590—42no—
44C2×Dic166—21no—
45C2×Dic166—42no—
46C2×Dic166C2312yes—
47C2×Dic166C2311yescharacteristic
48C2×D332C2311yescharacteristic
49C2×C22⋊D83—21no—
50C2×C22⋊D83—21no—
51C2×C22⋊D83—21no—
52C22×Dic83—21no—
53C22×Dic83—21no—
54C2×C4×D83—21no—
55Dic332—42no—
56#13050579—42no—
57C4⋊Dic83—21no—
58C4⋊Dic83C2311yescharacteristic
59C4⋊Dic83C2311yescharacteristic
60C4⋊Dic83C2312yes—
61Dic83⋊C4—22no—
62Dic83⋊C4—22no—
63Dic83⋊C4—21no—
64Dic83⋊C4—21no—
65C22⋊Dic83—21no—
66C22⋊Dic83—21no—
67C22⋊Dic83—21no—
68C22⋊Dic83—22no—
69C22⋊Dic83—22no—
70(C2×D83)⋊C4—21no—
71(C2×D83)⋊C4—21no—
72C4×Dic83—21no—
73C24×C83D411yescharacteristic
74C22×C4×C83—42no—
75C22×C4×C83D411yescharacteristic
76C22×C4×C83D411yescharacteristic
77C22×C4×C83D411yescharacteristic
78C22×C4×C83C2311yescharacteristic
79C22×C4×C83C2311yescharacteristic
80C22×C4×C83C2311yescharacteristic
81C2×C8×C83—42no—
82C2×C8×C83C2312yes—
83C2×C8×C83C2312yes—
84C23×C83—22no—
85C23×C83—22no—
86C23×C83—41no—
87C23×C83C2×D411yescharacteristic
88C23×C83C2×D411yescharacteristic
89C23×C83C2×D411yescharacteristic
90C22×D83—21no—
91C2×C4×C83—22no—
92C2×C4×C83—22no—
93C2×C4×C83—22no—
94C2×C4×C83—41no—
95C2×C4×C83—41no—
96C2×C4×C83—41no—
97C2×C4×C83C2×D411yescharacteristic
98C2×C4×C83C2×D411yescharacteristic
99C2×C4×C83C2×D411yescharacteristic
100C2×C4×C83C2411yescharacteristic
101C22⋊D83—42no—
102C22⋊D83—42no—
103C22⋊D83—42no—
104C2×Dic83—21no—
105C2×Dic83—21no—
106C2×Dic83—22no—
107C2×Dic83—21no—
108C2×Dic83—22no—
109C2×Dic83—42no—
110C2×Dic83—42no—
111D332—22no—
112C4×D83—42no—
113Dic166—22no—
114Dic166—22no—
115Dic166—22no—
116Dic166—42no—
117C8×C83—22no—
118C8×C83—22no—
119C22×C83—21no—
120C22×C83—21no—
121C22×C83—21no—
122C22×C83—21no—
123C22×C83—21no—
124C22×C83—21no—
125C22×C83—21no—
126C22×C83—44no—
127C22×C83#13512yes—
128C22×C83#13712yes—
129C22×C83C22×D411yescharacteristic
130C2×D83—41no—
131C2×D83—41no—
132C2×D83—41no—
133C4×C83—21no—
134C4×C83—21no—
135C4×C83—21no—
136C4×C83H(5,𝔽2)12yes—
137Dic83—42no—
138Dic83—41no—
139Dic83—42no—
140Dic83—41no—
141Dic83—41no—
142C2×C83—22no—
143C2×C83—22no—
144C2×C83—41no—
145C2×C83#53411yescharacteristic
146C2×C83C2×#13511yescharacteristic
147C2×C83C2×#13711yescharacteristic
148D83—81no—
149#3190—831nomaximal, Sylow
150C83#319011yescharacteristic, complemented, Sylow
151#525—831no—
152C2×C22⋊Q8—831no—
153#485—832no—
154#483—831no—
155#480—831no—
156#467—832no—
157#465—832no—
158#452—832no—
159#451—831no—
160#448—831no—
161C2×C22⋊C8D8311yescharacteristic, complemented
162C2×#41—831no—
163C22×Q8—831no—
164C23×C4C2×D8311yescharacteristic
165C2×Q16—832no—
166C2×SD16—832no—
167C22×C8C2×D8312yes—
168#123—1661no—
169C22⋊Q8—832no—
170C22⋊Q8—832no—
171C22⋊Q8—831no—
172C22⋊Q8—831no—
173C22⋊Q8—1661no—
174C22⋊Q8—1661no—
175#121—831no—
176#121—831no—
177#120—1661no—
178C4×D4—1661no—
179C4×D4—1661no—
180#117—831no—
181C2×C4⋊C4—831no—
182C2×C4⋊C4—1661no—
183C2×C22⋊C4—1661no—
184#107—832no—
185#106—832no—
186#103—832no—
187#103—832no—
188#103—832no—
189#102—832no—
190C22⋊C8C2×D8311yescharacteristic
191C22⋊C8C2×D8311yescharacteristic
192C22⋊C8C2×D8312yes—
193C24D33211yescharacteristic
194#41—3322no—
195C2×Q8—3322no—
196C2×Q8—832no—
197C2×Q8—831no—
198C2×Q8—1661no—
199C2×D4—831no—
200C2×D4—1661no—
201C2×D4—1661no—
202C2×D4—1661no—
203C22×C4—42no—
204C22×C4—1661no—
205C22×C4—1661no—
206C22×C4—1661no—
207C22×C4D33211yescharacteristic
208C22×C4D33211yescharacteristic
209C22×C4D33211yescharacteristic
210C22×C4C22×D8311yescharacteristic
211C22×C4C22×D8311yescharacteristic
212C22×C4C22×D8311yescharacteristic
213Q16—3322no—
214SD16—3322no—
215C2×C8—42no—
216C2×C8C22×D8312yes—
217C2×C8C22×D8312yes—
218C4⋊C4—832no—
219C4⋊C4—831no—
220C4⋊C4—831no—
221C4⋊C4—1661no—
222C4⋊C4—1662no—
223C4⋊C4—1662no—
224C4⋊C4—1661no—
225C4⋊C4—1661no—
226C22⋊C4—1662no—
227C22⋊C4—1662no—
228C22⋊C4—1661no—
229C22⋊C4—1661no—
230C22⋊C4—1661no—
231C22⋊C4—1661no—
232C22⋊C4—1661no—
233C42—1661no—
234C23—22no—
235C23—22no—
236C23—41no—
237C23—1661no—
238C23C2×D33211yescharacteristic
239C23C2×D33211yescharacteristic
240C23C2×D33211yescharacteristic
241Q8—3322no—
242Q8—1662no—
243Q8—1662no—
244Q8—1662no—
245D4—3322no—
246D4—3322no—
247D4—3322no—
248D4—1662no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—3322no—
256C2×C4—3322no—
257C2×C4—3322no—
258C2×C4—1661no—
259C2×C4—1661no—
260C2×C4—1662no—
261C2×C4—1661no—
262C2×C4—1662no—
263C2×C4C2×D33211yescharacteristic
264C2×C4C2×D33211yescharacteristic
265C2×C4C2×D33211yescharacteristic
266C2×C4C23×D8311yescharacteristic
267C8—22no—
268C8—22no—
269C22—21no—
270C22—21no—
271C22—21no—
272C22—21no—
273C22—21no—
274C22—21no—
275C22—21no—
276C22—44no—
277C22—3321no—
278C22—3321no—
279C22—3321no—
280C22#1458181412yes—
281C22#1458182512yes—
282C22C22×D33211yescharacteristic
283C4—21no—
284C4—21no—
285C4—21no—
286C4—3322no—
287C4—3321no—
288C4—3321no—
289C4—3321no—
290C4—3322no—
291C4#1458183712yes—
292C2—22no—
293C2—22no—
294C2—41no—
295C2—6641no—
296C2#1912826311yescharacteristic
297C2C2×#1458181411yescharacteristic
298C2C2×#1458182511yescharacteristic
subgroupG/H
centerC22C22×D332
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
FittingC2×C22⋊C8×C83C2
socleC22×C83C22×D4
prime radical#19128212C2
hypercenterC2×C22⋊C8D83
hypocenterC83#3190
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×C22⋊C8

Complements

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

NH ≅ G/Ntype
2. C2×#14581771C2semidirect
3. #19128043C2semidirect
4. #19128032C2semidirect
5. #19128015C2semidirect
6. #19127942C2semidirect
7. #19127930C2semidirect
8. #19127865C2semidirect
9. #19127859C2semidirect
10. #19127842C2semidirect
11. C2×C22⋊C8×C83C2semidirect
14. C2×Dic332C22semidirect
15. C2×#13050579C22semidirect
32. #14581726C22semidirect
33. #14581724C22semidirect
34. #14581717C22semidirect
35. #14581717C22semidirect
36. #14581717C22semidirect
37. #14581714C22semidirect
150. C83#3190semidirect
161. C2×C22⋊C8D83semidirect

Maximal subgroups

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

subgroupindex
#191282122
C2×#145817712
#191280432
#191280322
#191280152
#191279422
#191279302
#191278652
#191278592
#191278422
C2×C22⋊C8×C832
#319083