Finite Group Databasebeta

← #24918241

Subgroups — #24918241

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#19128211C211yesmaximal, characteristic
2C2×#14581768C211yesmaximal, characteristic, complemented
3#19128040C212yesmaximal, complemented
4#19128032C211yesmaximal, characteristic, complemented
5#19128012C211yesmaximal, characteristic, complemented
6#19127941C212yesmaximal, complemented
7#19127927C212yesmaximal, complemented
8#19127860C212yesmaximal, complemented
9#19127857C211yesmaximal, characteristic, complemented
10C22⋊C8×D83C211yesmaximal, characteristic, complemented
11#448×C83C211yesmaximal, characteristic, complemented
12C2×#13050589C2211yescharacteristic
13C22×C4×D83C2211yescharacteristic
14C2×#13050580C2212yescomplemented
15C2×C8×D83C2212yes—
16#14581774—21no—
17#14581772C2211yescharacteristic
18#14581771C2211yescharacteristic
19#14581768—21no—
20#14581768—21no—
21#14581768C2212yes—
22#14581768C2212yes—
23#14581763C2211yescharacteristic
24#14581760—21no—
25D4×Dic83—21no—
26(C22⋊D83)⋊C4—21no—
27#14581750C2211yescharacteristic
28C2×C4⋊Dic83C2211yescharacteristic
29C2×Dic83⋊C4—21no—
30C2×(C2×D83)⋊C4—21no—
31#14581725C2212yescomplemented
32#14581724C2212yescomplemented
33#14581717C2212yescomplemented
34#14581716C2212yescomplemented
35#14581713C2212yescomplemented
36C22⋊(C83⋊C8)C2211yescharacteristic
37(C2×D83)⋊C8C2212yes—
38C22×Q8×C83C2211yescharacteristic
39C2×SD16×C83C2212yescomplemented
40#121×C83C2211yescharacteristic
41#103×C83C2212yescomplemented
42C22⋊C8×C83C2211yescharacteristic
43C23×D83D411yescharacteristic
44#13050589—42no—
45C2×Dic166C2311yescharacteristic
46C2×C22⋊D83—21no—
47C2×C22⋊D83—21no—
48C22×Dic83—21no—
49C22×Dic83—21no—
50C22×Dic83D411yescharacteristic
51C2×C4×D83—42no—
52C2×C4×D83D411yescharacteristic
53C2×C4×D83D411yescharacteristic
54C2×C4×D83C2311yescharacteristic
55C2×C4×D83C2311yescharacteristic
56#13050580—42no—
57C2×C83⋊C8C2312yes—
58C8×D83—42no—
59C4⋊Dic83—21no—
60C4⋊Dic83—21no—
61C4⋊Dic83C2311yescharacteristic
62C4⋊Dic83C2312yes—
63Dic83⋊C4—21no—
64Dic83⋊C4—22no—
65Dic83⋊C4—22no—
66C22⋊Dic83—21no—
67C22⋊Dic83—21no—
68(C2×D83)⋊C4—21no—
69(C2×D83)⋊C4—21no—
70(C2×D83)⋊C4—22no—
71(C2×D83)⋊C4—22no—
72C4×Dic83—21no—
73C2×Q8×C83—21no—
74C2×Q8×C83—42no—
75C2×Q8×C83C2312yes—
76C2×D4×C83—21no—
77C2×D4×C83C2311yescharacteristic
78C22×C4×C83—21no—
79C22×C4×C83C2311yescharacteristic
80SD16×C83—42no—
81C2×C8×C83C2312yes—
82C4⋊C4×C83C2311yescharacteristic
83C22⋊C4×C83—21no—
84C23×C83—21no—
85C23×C83C2×D411yescharacteristic
86C22×D83—22no—
87C22×D83—22no—
88C22×D83—41no—
89C22×D83C2×D411yescharacteristic
90C22×D83C2×D411yescharacteristic
91Q8×C83—22no—
92Q8×C83—22no—
93Q8×C83—42no—
94D4×C83—22no—
95D4×C83—42no—
96C2×C4×C83—22no—
97C2×C4×C83—21no—
98C2×C4×C83—22no—
99C2×C4×C83—42no—
100C2×C4×C83C2×D411yescharacteristic
101C2×C4×C83C2411yescharacteristic
102C22⋊D83—42no—
103C22⋊D83—42no—
104C2×Dic83—21no—
105C2×Dic83—22no—
106C2×Dic83—21no—
107C2×Dic83—41no—
108C2×Dic83—42no—
109C2×Dic83—42no—
110C2×Dic83C2×D411yescharacteristic
111C2×Dic83C2×D411yescharacteristic
112C4×D83—22no—
113C4×D83—22no—
114C4×D83—41no—
115C4×D83—41no—
116Dic166—22no—
117C8×C83—22no—
118C83⋊C8—22no—
119C22×C83—21no—
120C22×C83—21no—
121C22×C83—41no—
122C22×C83—41no—
123C22×C83—41no—
124C22×C83#13512yes—
125C22×C83C22×D411yescharacteristic
126C2×D83—21no—
127C2×D83—21no—
128C2×D83—21no—
129C2×D83—21no—
130C2×D83—21no—
131C2×D83—44no—
132C2×D83#13712yes—
133C4×C83—21no—
134C4×C83—42no—
135C4×C83—41no—
136C4×C83H(5,𝔽2)12yes—
137Dic83—21no—
138Dic83—21no—
139Dic83—41no—
140Dic83—42no—
141Dic83—41no—
142C2×C83—22no—
143C2×C83—81no—
144C2×C83#53411yescharacteristic
145C2×C83C2×#13511yescharacteristic
146C2×C83C2×#13711yescharacteristic
147D83—22no—
148D83—41no—
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#448D8311yescharacteristic, complemented
161C2×C22⋊C8—831no—
162C2×#41—831no—
163C22×Q8C2×D8311yescharacteristic
164C23×C4—831no—
165C2×Q16—832no—
166C2×SD16C2×D8312yescomplemented
167C22×C8—832no—
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#121C2×D8311yescharacteristic
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#103C2×D8312yescomplemented
189#102—832no—
190C22⋊C8—831no—
191C22⋊C8—832no—
192C22⋊C8C2×D8311yescharacteristic
193C24—831no—
194#41—3322no—
195C2×Q8—21no—
196C2×Q8—42no—
197C2×Q8—831no—
198C2×Q8C22×D8312yes—
199C2×D4—21no—
200C2×D4—1661no—
201C2×D4—1661no—
202C2×D4C22×D8311yescharacteristic
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—42no—
215C2×C8—3322no—
216C2×C8—832no—
217C2×C8C22×D8312yes—
218C4⋊C4—832no—
219C4⋊C4—831no—
220C4⋊C4—1662no—
221C4⋊C4—1662no—
222C4⋊C4—1661no—
223C4⋊C4—1661no—
224C4⋊C4—1661no—
225C4⋊C4C22×D8311yescharacteristic
226C22⋊C4—21no—
227C22⋊C4—1662no—
228C22⋊C4—1662no—
229C22⋊C4—1661no—
230C22⋊C4—1661no—
231C22⋊C4—1661no—
232C22⋊C4—1661no—
233C42—1661no—
234C23—21no—
235C23—3321no—
236C23—831no—
237C23—831no—
238C23—1662no—
239C23—1662no—
240C23D4×D8311yescharacteristic
241Q8—22no—
242Q8—22no—
243Q8—42no—
244Q8—1662no—
245D4—22no—
246D4—42no—
247D4—3322no—
248D4—3322no—
249C2×C4—22no—
250C2×C4—21no—
251C2×C4—22no—
252C2×C4—42no—
253C2×C4—3321no—
254C2×C4—3321no—
255C2×C4—3322no—
256C2×C4—3322no—
257C2×C4—3321no—
258C2×C4—831no—
259C2×C4—831no—
260C2×C4—1661no—
261C2×C4—1662no—
262C2×C4—1661no—
263C2×C4—1662no—
264C2×C4—1662no—
265C2×C4D4×D8311yescharacteristic
266C2×C4C23×D8311yescharacteristic
267C8—22no—
268C8—1662no—
269C22—21no—
270C22—21no—
271C22—41no—
272C22—41no—
273C22—41no—
274C22—3324no—
275C22—832no—
276C22—1661no—
277C22—1661no—
278C22—1661no—
279C22—1661no—
280C22—1661no—
281C22#1458181312yes—
282C22C2×D4×D8311yescharacteristic
283C4—21no—
284C4—41no—
285C4—42no—
286C4—3322no—
287C4—3321no—
288C4—3321no—
289C4—1661no—
290C4—1661no—
291C4#1458183712yes—
292C2—22no—
293C2—81no—
294C2—3321no—
295C2—1662no—
296C2#1912826111yescharacteristic
297C2C2×#1458181311yescharacteristic
298C2#137×D8311yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#448×C83C2
socleC22×C83C22×D4
prime radical#19128211C2
hypercenter#448D83
hypocenterC83#3190
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ #448

Complements

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

NH ≅ G/Ntype
2. C2×#14581768C2semidirect
3. #19128040C2semidirect
4. #19128032C2semidirect
5. #19128012C2semidirect
6. #19127941C2semidirect
7. #19127927C2semidirect
8. #19127860C2semidirect
9. #19127857C2semidirect
10. C22⋊C8×D83C2semidirect
11. #448×C83C2semidirect
14. C2×#13050580C22semidirect
31. #14581725C22semidirect
32. #14581724C22semidirect
33. #14581717C22semidirect
34. #14581716C22semidirect
35. #14581713C22semidirect
39. C2×SD16×C83C22semidirect
41. #103×C83C22semidirect
150. C83#3190semidirect
160. #448D83semidirect
166. C2×SD16C2×D83semidirect
188. #103C2×D83semidirect

Maximal subgroups

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

subgroupindex
#191282112
C2×#145817682
#191280402
#191280322
#191280122
#191279412
#191279272
#191278602
#191278572
C22⋊C8×D832
#448×C832
#319083