Finite Group Databasebeta

← #24911734

Subgroups — #24911734

Order 10624.

672 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 672 conjugacy classes of subgroups — too many to draw. Only its 149 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×#14581748C211yesmaximal, characteristic, complemented
2C22×C22⋊Dic83C211yesmaximal, characteristic
3#19127637C212yesmaximal, complemented
4#19127629C212yesmaximal, complemented
5#19127595C212yesmaximal, complemented
6#19127594C212yesmaximal, complemented
7#19127576C211yesmaximal, characteristic, complemented
8#19127573C211yesmaximal, characteristic, complemented
9#19127562C211yesmaximal, characteristic, complemented
10#19127557C211yesmaximal, characteristic
11#379×C83C211yesmaximal, characteristic
12C23×Dic83C2211yescharacteristic
13C23×Dic83C2211yescharacteristic
14#14581748—24no—
15C2×C4⋊Dic83C2212yescomplemented
16C2×Dic83⋊C4C2211yescharacteristic, complemented
17C2×Dic83⋊C4C2211yescharacteristic, complemented
18C2×Dic83⋊C4C2212yescomplemented
19C2×C22⋊Dic83—24no—
20C2×C22⋊Dic83C2211yescharacteristic
21C2×C22⋊Dic83C2211yescharacteristic
22C2×C22⋊Dic83C2212yes—
23C2×C22⋊Dic83C2212yes—
24C2×C22⋊Dic83C2212yes—
25C2×C22⋊Dic83C2212yes—
26C2×C4×Dic83C2211yescharacteristic, complemented
27C2×C4×Dic83C2211yescharacteristic, complemented
28C83⋊#95C2212yescomplemented
29C83⋊#95C2212yescomplemented
30C83⋊#95C2212yescomplemented
31C83⋊#95C2212yescomplemented
32C25×C83C2211yescharacteristic
33C2×C22⋊C4×C83C2211yescharacteristic
34C2×C22⋊C4×C83C2211yescharacteristic
35C2×C22⋊C4×C83C2212yes—
36C2×C22⋊C4×C83C2212yes—
37C22×Dic83—22no—
38C22×Dic83—22no—
39C22×Dic83—22no—
40C22×Dic83—42no—
41C22×Dic83D414yes—
42C22×Dic83C2312yes—
43C22×Dic83C2312yes—
44C22×Dic83C2311yescharacteristic
45C22×Dic83C2311yescharacteristic
46C22×Dic83C2311yescharacteristic
47C22×Dic83C2311yescharacteristic
48C4⋊Dic83—42no—
49Dic83⋊C4—22no—
50Dic83⋊C4—22no—
51Dic83⋊C4—42no—
52C22⋊Dic83—22no—
53C22⋊Dic83—24no—
54C22⋊Dic83—22no—
55C22⋊Dic83—24no—
56C22⋊Dic83—24no—
57C22⋊Dic83—42no—
58C4×Dic83—22no—
59C4×Dic83—22no—
60C24×C83—22no—
61C24×C83—24no—
62C24×C83—22no—
63C24×C83—22no—
64C24×C83—42no—
65C24×C83C2312yes—
66C24×C83C2311yescharacteristic
67C22×C4×C83C2312yes—
68C22×C4×C83C2311yescharacteristic
69C22×C4×C83C2311yescharacteristic
70C22⋊C4×C83—22no—
71C22⋊C4×C83—24no—
72C22⋊C4×C83—22no—
73C22⋊C4×C83—42no—
74C23×C83—22no—
75C23×C83—22no—
76C23×C83—28no—
77C23×C83—22no—
78C23×C83—22no—
79C23×C83—22no—
80C23×C83—22no—
81C23×C83—22no—
82C23×C83—22no—
83C23×C83—48no—
84C23×C83—48no—
85C23×C83—44no—
86C23×C83—44no—
87C23×C83C2×D412yes—
88C23×C83#4114yes—
89C23×C83#4112yes—
90C23×C83#4112yes—
91C23×C83C2411yescharacteristic
92C2×C4×C83—21no—
93C2×C4×C83—21no—
94C2×C4×C83—22no—
95C2×C4×C83—22no—
96C2×C4×C83—22no—
97C2×C4×C83—22no—
98C2×C4×C83—22no—
99C2×Dic83—28no—
100C2×Dic83—21no—
101C2×Dic83—21no—
102C2×Dic83—21no—
103C2×Dic83—21no—
104C2×Dic83—21no—
105C2×Dic83—21no—
106C2×Dic83—22no—
107C2×Dic83—22no—
108C2×Dic83—22no—
109C2×Dic83—22no—
110C2×Dic83—22no—
111C2×Dic83—22no—
112C2×Dic83—22no—
113C2×Dic83—22no—
114C2×Dic83—44no—
115C2×Dic83C2×D412yes—
116C2×Dic83C2×D412yes—
117C22×C83—22no—
118C22×C83—22no—
119C22×C83—24no—
120C22×C83—24no—
121C22×C83—24no—
122C22×C83—24no—
123C22×C83—24no—
124C22×C83—416no—
125C22×C83—42no—
126C22×C83—42no—
127C22×C83—42no—
128C22×C83—42no—
129C22×C83#12314yes—
130C22×C83C22×D411yescharacteristic
131C22×C83C2×#4111yescharacteristic
132C22×C83C2×#4112yes—
133C22×C83C2×#4111yescharacteristic
134C22×C83H(5,𝔽2)12yes—
135C4×C83—41no—
136C4×C83—41no—
137C4×C83—42no—
138Dic83—22no—
139Dic83—22no—
140Dic83—41no—
141Dic83—41no—
142Dic83—42no—
143Dic83—42no—
144C2×C83—24no—
145C2×C83—44no—
146C2×C83C2×#12311yescharacteristic
147C2×C83#52511yescharacteristic
148C2×C83#53511yescharacteristic
149C2×C83#55112yes—
150C2×C83#55112yes—
151#2482—831nomaximal, Sylow
152C83#248211yescharacteristic, complemented, Sylow
153C2×#117—831no—
154C22×C22⋊C4—831no—
155#399—832no—
156#397—832no—
157#388—832no—
158#388—832no—
159#385—831no—
160#385—831no—
161#381—831no—
162#379—831no—
163#379D8311yescharacteristic
164C25C2×D8311yescharacteristic
165C23×C4—831no—
166C23×C4—831no—
167#117—1664no—
168C2×C4⋊C4—832no—
169C2×C4⋊C4—832no—
170C2×C4⋊C4—831no—
171C2×C4⋊C4—831no—
172C2×C22⋊C4—832no—
173C2×C22⋊C4—832no—
174C2×C22⋊C4—832no—
175C2×C22⋊C4—831no—
176C2×C22⋊C4—831no—
177C2×C22⋊C4—832no—
178C2×C22⋊C4—1664no—
179C2×C22⋊C4C2×D8311yescharacteristic
180C2×C22⋊C4C2×D8311yescharacteristic
181C2×C22⋊C4C2×D8312yes—
182C2×C22⋊C4C2×D8312yes—
183C2×C42—831no—
184C2×C42—831no—
185#95—832no—
186#95—832no—
187#95—832no—
188#95—832no—
189C24—22no—
190C24—24no—
191C24—22no—
192C24—22no—
193C24—42no—
194C24C22×D8312yes—
195C24C22×D8311yescharacteristic
196C22×C4—3322no—
197C22×C4—834no—
198C22×C4—831no—
199C22×C4—831no—
200C22×C4—832no—
201C22×C4—831no—
202C22×C4—832no—
203C22×C4—831no—
204C22×C4—1662no—
205C22×C4—1662no—
206C22×C4—1662no—
207C22×C4C22×D8312yes—
208C22×C4C22×D8311yescharacteristic
209C22×C4C22×D8311yescharacteristic
210C4⋊C4—3322no—
211C4⋊C4—3322no—
212C4⋊C4—1662no—
213C4⋊C4—1662no—
214C22⋊C4—22no—
215C22⋊C4—24no—
216C22⋊C4—22no—
217C22⋊C4—42no—
218C22⋊C4—3322no—
219C22⋊C4—1664no—
220C22⋊C4—1664no—
221C22⋊C4—1662no—
222C22⋊C4—1664no—
223C22⋊C4—1662no—
224C42—1662no—
225C42—1662no—
226C23—22no—
227C23—22no—
228C23—28no—
229C23—22no—
230C23—22no—
231C23—22no—
232C23—22no—
233C23—22no—
234C23—22no—
235C23—48no—
236C23—48no—
237C23—44no—
238C23—44no—
239C23D4×D8312yes—
240C23#1305058914yes—
241C23#1305058912yes—
242C23#1305059012yes—
243C23C23×D8311yescharacteristic
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—3324no—
252C2×C4—832no—
253C2×C4—832no—
254C2×C4—1661no—
255C2×C4—1661no—
256C2×C4—1662no—
257C2×C4—1662no—
258C2×C4—1662no—
259C2×C4—1662no—
260C2×C4—1662no—
261C2×C4—1662no—
262C2×C4—1662no—
263C2×C4—1662no—
264C2×C4—1668no—
265C2×C4—1661no—
266C2×C4—1661no—
267C2×C4—1661no—
268C2×C4—1661no—
269C22—24no—
270C22—24no—
271C22—22no—
272C22—22no—
273C22—24no—
274C22—24no—
275C22—24no—
276C22—416no—
277C22—42no—
278C22—42no—
279C22—42no—
280C22—42no—
281C22#1458177414yes—
282C22C2×D4×D8311yescharacteristic
283C22C2×#1305058912yes—
284C22C2×#1305058911yescharacteristic
285C22C2×#1305059011yescharacteristic
286C22#1458183712yes—
287C4—42no—
288C4—41no—
289C4—41no—
290C4—3322no—
291C4—3321no—
292C4—3322no—
293C4—3321no—
294C4—1662no—
295C4—1662no—
296C2—24no—
297C2—44no—
298C2C2×#1458177411yescharacteristic
299C2#1912820711yescharacteristic
300C2#1912826611yescharacteristic
301C2#1912837712yes—
302C2#1912838212yes—
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
Fitting#379×C83C2
prime radicalC25×C83C22
hypercenter#379D83
hypocenterC83#2482
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ #379

Complements

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

NH ≅ G/Ntype
1. C2×#14581748C2semidirect
3. #19127637C2semidirect
4. #19127629C2semidirect
5. #19127595C2semidirect
6. #19127594C2semidirect
7. #19127576C2semidirect
8. #19127573C2semidirect
9. #19127562C2semidirect
15. C2×C4⋊Dic83C22semidirect
16. C2×Dic83⋊C4C22semidirect
17. C2×Dic83⋊C4C22semidirect
18. C2×Dic83⋊C4C22semidirect
26. C2×C4×Dic83C22semidirect
27. C2×C4×Dic83C22semidirect
28. C83⋊#95C22semidirect
29. C83⋊#95C22semidirect
30. C83⋊#95C22semidirect
31. C83⋊#95C22semidirect
152. C83#2482semidirect

Maximal subgroups

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

subgroupindex
C2×#145817482
C22×C22⋊Dic832
#191276372
#191276292
#191275952
#191275942
#191275762
#191275732
#191275622
#191275572
#379×C832
#248283