Finite Group Databasebeta

← #24915751

Subgroups — #24915751

Order 10624.

1068 conjugacy classes of subgroups; only the 201 normal subgroups are shown.

normal subgroups only

This group has 1068 conjugacy classes of subgroups — too many to draw. Only its 201 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
1C23×C4×D83C211yesmaximal, characteristic
2C2×#14581772C212yesmaximal, complemented
3C2×#14581771C211yesmaximal, characteristic, complemented
4#19127576C212yesmaximal, complemented
5#19127575C214yesmaximal, complemented
6#19127558C212yesmaximal, complemented
7#19127556C212yesmaximal
8#380×C83C211yesmaximal, characteristic, complemented
9C24×D83C2211yescharacteristic
10C22×Dic166C2211yescharacteristic, complemented
11C23×Dic83C2211yescharacteristic
12C22×C4×D83—28no—
13C22×C4×D83—24no—
14C22×C4×D83C2212yes—
15C22×C4×D83C2212yes—
16#14581772—28no—
17#14581771—24no—
18C2×C4⋊Dic83C2211yescharacteristic, complemented
19C2×C4⋊Dic83C2214yescomplemented
20C2×Dic83⋊C4C2212yescomplemented
21C2×C22⋊Dic83C2212yes—
22C2×(C2×D83)⋊C4C2214yes—
23C2×(C2×D83)⋊C4C2214yes—
24C83⋊#95C2214yescomplemented
25C23×C4×C83C2211yescharacteristic
26C2×C4⋊C4×C83C2212yescomplemented
27C2×C22⋊C4×C83C2212yes—
28#95×C83C2212yescomplemented
29C23×D83—22no—
30C23×D83—22no—
31C23×D83—24no—
32C23×D83—24no—
33C23×D83Q814yescomplemented
34C23×D83C2312yes—
35C2×Dic166—22no—
36C2×Dic166—24no—
37C22×Dic83—22no—
38C22×Dic83—24no—
39C22×Dic83C2312yes—
40C22×Dic83C2312yes—
41C22×Dic83C2312yes—
42C2×C4×D83—232no—
43C2×C4×D83—24no—
44C2×C4×D83—24no—
45C2×C4×D83—24no—
46C2×C4×D83—24no—
47C2×C4×D83—24no—
48C2×C4×D83D418yes—
49C4⋊Dic83—22no—
50C4⋊Dic83—28no—
51Dic83⋊C4—24no—
52C22⋊Dic83—24no—
53(C2×D83)⋊C4—28no—
54(C2×D83)⋊C4—28no—
55C24×C83C2311yescharacteristic
56C22×C4×C83—24no—
57C22×C4×C83D414yes—
58C22×C4×C83C2314yes—
59C22×C4×C83C2311yescharacteristic
60C22×C4×C83C2311yescharacteristic
61C4⋊C4×C83—24no—
62C22⋊C4×C83—24no—
63C23×C83—22no—
64C23×C83—22no—
65C23×C83C2×D412yes—
66C23×C83C2×Q812yes—
67C23×C83#4112yes—
68C23×C83C2411yescharacteristic
69C22×D83—28no—
70C22×D83—216no—
71C22×D83—28no—
72C22×D83—28no—
73C22×D83—22no—
74C22×D83—22no—
75C22×D83—22no—
76C22×D83—22no—
77C22×D83—216no—
78C22×D83C2×D414yes—
79C22×D83C2×Q814yes—
80C22×D83#4114yes—
81C2×C4×C83—22no—
82C2×C4×C83—28no—
83C2×C4×C83—24no—
84C2×C4×C83—24no—
85C2×C4×C83—24no—
86C2×C4×C83C2×D414yes—
87C2×C4×C83C2×D412yes—
88C2×C4×C83C2×D412yes—
89C2×Dic83—28no—
90C2×Dic83—22no—
91C2×Dic83—22no—
92C2×Dic83—24no—
93C2×Dic83—22no—
94C2×Dic83—22no—
95C2×Dic83—24no—
96C2×Dic83C2×D414yes—
97C4×D83—216no—
98C4×D83—216no—
99Dic166—44no—
100C22×C83—22no—
101C22×C83—24no—
102C22×C83—22no—
103C22×C83—24no—
104C22×C83C22⋊Q814yescomplemented
105C22×C83C22×D412yes—
106C22×C83C22×D411yescharacteristic
107C22×C83C22×Q811yescharacteristic
108C22×C83C2×#4112yes—
109C22×C83C2×#4111yescharacteristic
110C2×D83—232no—
111C2×D83—24no—
112C2×D83—24no—
113C2×D83—24no—
114C2×D83—24no—
115C2×D83—24no—
116C2×D83—24no—
117C2×D83C22⋊Q818yescomplemented
118C4×C83—22no—
119C4×C83—44no—
120C4×C83#12014yes—
121Dic83—24no—
122Dic83—42no—
123Dic83—42no—
124C2×C83—24no—
125C2×C83C2×#12011yescharacteristic
126C2×C83C2×C22⋊Q812yes—
127C2×C83C2×C22⋊Q811yescharacteristic
128C2×C83#52512yes—
129C2×C83#52511yescharacteristic
130D83—28no—
131#2848—831nomaximal, Sylow
132C83#284811yescharacteristic, complemented, Sylow
133C24×C4—831no—
134C2×C22⋊Q8—831no—
135C2×C22⋊Q8—832no—
136#385—832no—
137#385—834no—
138#380—832no—
139#380D8311yescharacteristic, complemented
140#379—832no—
141C25—831no—
142C22×Q8—831no—
143C23×C4—832no—
144C23×C4—831no—
145C23×C4—832no—
146C23×C4—1668no—
147C23×C4—1664no—
148C23×C4C2×D8311yescharacteristic
149C22⋊Q8—1668no—
150C22⋊Q8—1664no—
151C2×C4⋊C4—831no—
152C2×C4⋊C4—832no—
153C2×C4⋊C4—834no—
154C2×C4⋊C4C2×D8312yescomplemented
155C2×C22⋊C4—832no—
156C2×C22⋊C4—834no—
157C2×C22⋊C4—834no—
158C2×C22⋊C4C2×D8312yes—
159#95—834no—
160#95C2×D8312yescomplemented
161C24—832no—
162C24—834no—
163C24—1664no—
164C24—1664no—
165C24—1662no—
166C24—1662no—
167C24C22×D8311yescharacteristic
168C2×Q8—1664no—
169C2×Q8—1662no—
170C22×C4—24no—
171C22×C4—832no—
172C22×C4—838no—
173C22×C4—832no—
174C22×C4—832no—
175C22×C4—1664no—
176C22×C4—1664no—
177C22×C4—16632no—
178C22×C4—1664no—
179C22×C4—1662no—
180C22×C4—1664no—
181C22×C4—1664no—
182C22×C4—1664no—
183C22×C4D33214yes—
184C22×C4C22×D8314yes—
185C22×C4C22×D8311yescharacteristic
186C22×C4C22×D8311yescharacteristic
187C4⋊C4—24no—
188C4⋊C4—1668no—
189C4⋊C4—1664no—
190C4⋊C4—1662no—
191C22⋊C4—24no—
192C22⋊C4—1668no—
193C22⋊C4—1668no—
194C22⋊C4—1664no—
195C23—22no—
196C23—22no—
197C23—834no—
198C23—834no—
199C23—834no—
200C23—16616no—
201C23—1668no—
202C23—16616no—
203C23—1668no—
204C23—1668no—
205C23—1662no—
206C23—1662no—
207C23—1662no—
208C23—1662no—
209C23C2×D33212yes—
210C23Q8×D8312yes—
211C23#1305058912yes—
212C23C23×D8311yescharacteristic
213Q8—3324no—
214C2×C4—22no—
215C2×C4—28no—
216C2×C4—24no—
217C2×C4—24no—
218C2×C4—24no—
219C2×C4—834no—
220C2×C4—16616no—
221C2×C4—1664no—
222C2×C4—1662no—
223C2×C4—1662no—
224C2×C4—1664no—
225C2×C4—1668no—
226C2×C4—1662no—
227C2×C4—1662no—
228C2×C4—16616no—
229C2×C4D4×D8314yes—
230C2×C4C2×D33212yes—
231C2×C4C2×D33212yes—
232C22—22no—
233C22—24no—
234C22—24no—
235C22—22no—
236C22—838no—
237C22—16632no—
238C22—1664no—
239C22—1664no—
240C22—1664no—
241C22—1664no—
242C22—1664no—
243C22—1664no—
244C22#1458177214yescomplemented
245C22C2×D4×D8312yes—
246C22C22×D33211yescharacteristic
247C22C2×Q8×D8311yescharacteristic
248C22#41×D8312yes—
249C22C2×#1305058911yescharacteristic
250C4—22no—
251C4—44no—
252C4—3322no—
253C4—3322no—
254C4—1664no—
255C4#1458176214yes—
256C2—24no—
257C2—1668no—
258C2C2×#1458176211yescharacteristic
259C2C22⋊Q8×D8312yes—
260C2C2×#1458177211yescharacteristic
261C2#1912821112yes—
262C2#1912821311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
Fitting#380×C83C2
prime radicalC24×D83C22
hypercenter#380D83
hypocenterC83#2848
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ #380

Complements

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

NH ≅ G/Ntype
2. C2×#14581772C2semidirect
3. C2×#14581771C2semidirect
4. #19127576C2semidirect
5. #19127575C2semidirect
6. #19127558C2semidirect
8. #380×C83C2semidirect
10. C22×Dic166C22semidirect
18. C2×C4⋊Dic83C22semidirect
19. C2×C4⋊Dic83C22semidirect
20. C2×Dic83⋊C4C22semidirect
24. C83⋊#95C22semidirect
26. C2×C4⋊C4×C83C22semidirect
28. #95×C83C22semidirect
33. C23×D83Q8semidirect
104. C22×C83C22⋊Q8semidirect
117. C2×D83C22⋊Q8semidirect
132. C83#2848semidirect
139. #380D83semidirect
154. C2×C4⋊C4C2×D83semidirect
160. #95C2×D83semidirect
244. C22#14581772semidirect

Maximal subgroups

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

subgroupindex
C23×C4×D832
C2×#145817722
C2×#145817712
#191275762
#191275752
#191275582
#191275562
#380×C832
#284883