Finite Group Databasebeta

← #24910813

Subgroups — #24910813

Order 10624.

1232 conjugacy classes of subgroups; only the 205 normal subgroups are shown.

normal subgroups only

This group has 1232 conjugacy classes of subgroups — too many to draw. Only its 205 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×#14581763C212yesmaximal, complemented
2C2×#14581761C212yesmaximal, complemented
3C2×C4×C22⋊D83C211yesmaximal, characteristic, complemented
4#19127615C212yesmaximal, complemented
5#19127609C212yesmaximal, complemented
6#19127604C212yesmaximal, complemented
7#19127561C211yesmaximal, characteristic, complemented
8#19127557C212yesmaximal, complemented
9C23×D4×C83C211yesmaximal, characteristic, complemented
10C22×C22⋊D83C2211yescharacteristic
11C22×C22⋊D83C2214yescomplemented
12C22×C4×D83C2211yescharacteristic, complemented
13#14581763—28no—
14#14581761—28no—
15C4×C22⋊D83—24no—
16C2×C4⋊Dic83C2212yescomplemented
17C2×Dic83⋊C4C2211yescharacteristic, complemented
18C2×C22⋊Dic83C2211yescharacteristic
19C2×C22⋊Dic83C2212yes—
20C2×C22⋊Dic83C2214yescomplemented
21C2×C22⋊Dic83C2214yescomplemented
22C2×C22⋊Dic83C2214yescomplemented
23C2×(C2×D83)⋊C4C2211yescharacteristic, complemented
24C2×C4×Dic83C2211yescharacteristic, complemented
25C83⋊#95C2212yescomplemented
26C25×C83C2212yes—
27C22×D4×C83—28no—
28C22×D4×C83—24no—
29C22×D4×C83C2212yes—
30C22×D4×C83C2212yes—
31C23×C4×C83C2211yescharacteristic
32C23×D83C2311yescharacteristic
33C2×C22⋊D83—22no—
34C2×C22⋊D83—22no—
35C2×C22⋊D83—28no—
36C2×C22⋊D83—28no—
37C2×C22⋊D83—44no—
38C2×C22⋊D83D414yescomplemented
39C22×Dic83C2311yescharacteristic
40C22×Dic83C2314yes—
41C22×Dic83C2311yescharacteristic
42C22×Dic83C2311yescharacteristic
43C2×C4×D83—24no—
44C2×C4×D83—22no—
45C4⋊Dic83—24no—
46Dic83⋊C4—22no—
47C22⋊Dic83—24no—
48C22⋊Dic83—28no—
49C22⋊Dic83—22no—
50C22⋊Dic83—28no—
51C22⋊Dic83—44no—
52(C2×D83)⋊C4—22no—
53C4×Dic83—22no—
54C24×C83—28no—
55C24×C83—24no—
56C24×C83—24no—
57C24×C83—24no—
58C24×C83—44no—
59C24×C83C2314yes—
60C24×C83C2311yescharacteristic
61C2×D4×C83—232no—
62C2×D4×C83—28no—
63C2×D4×C83—24no—
64C2×D4×C83—24no—
65C2×D4×C83—24no—
66C2×D4×C83D418yescomplemented
67C22×C4×C83—24no—
68C22×C4×C83D414yescomplemented
69C22×C4×C83C2311yescharacteristic
70C22×C4×C83C2311yescharacteristic
71C23×C83—22no—
72C23×C83—22no—
73C23×C83—216no—
74C23×C83—24no—
75C23×C83—24no—
76C23×C83—24no—
77C23×C83—24no—
78C23×C83—24no—
79C23×C83—24no—
80C23×C83—416no—
81C23×C83—416no—
82C23×C83—48no—
83C23×C83—48no—
84C23×C83C2×D412yes—
85C23×C83C2×D418yes—
86C23×C83C2×D412yes—
87C23×C83#4112yes—
88C23×C83C2411yescharacteristic
89C22×D83—21no—
90C22×D83—22no—
91C22×D83—21no—
92C22×D83—22no—
93C22×D83C2×D412yescomplemented
94D4×C83—216no—
95D4×C83—216no—
96C2×C4×C83—22no—
97C2×C4×C83—28no—
98C2×C4×C83C2×D412yes—
99C2×C4×C83C2×D414yes—
100C2×C4×C83C2×D412yes—
101C22⋊D83—28no—
102C22⋊D83—416no—
103C2×Dic83—21no—
104C2×Dic83—21no—
105C2×Dic83—21no—
106C2×Dic83—22no—
107C2×Dic83—24no—
108C2×Dic83—21no—
109C2×Dic83—24no—
110C2×Dic83—22no—
111C2×Dic83—24no—
112C2×Dic83C2×D412yescomplemented
113C4×D83—44no—
114C22×C83—22no—
115C22×C83—22no—
116C22×C83—24no—
117C22×C83—24no—
118C22×C83—28no—
119C22×C83—28no—
120C22×C83—28no—
121C22×C83—432no—
122C22×C83—44no—
123C22×C83—44no—
124C22×C83—44no—
125C22×C83—44no—
126C22×C83#12114yescomplemented
127C22×C83C22×D411yescharacteristic
128C22×C83C22×D411yescharacteristic
129C22×C83C22×D412yes—
130C22×C83C2×#4111yescharacteristic
131C22×C83H(5,𝔽2)12yes—
132C2×D83—22no—
133C2×D83—24no—
134C2×D83—41no—
135C2×D83—41no—
136C2×D83—42no—
137C4×C83—22no—
138C4×C83#12014yescomplemented
139Dic83—22no—
140Dic83—41no—
141Dic83—41no—
142Dic83—44no—
143C2×C83—24no—
144C2×C83—48no—
145C2×C83C2×#12011yescharacteristic
146C2×C83C2×#12111yescharacteristic
147C2×C83#53511yescharacteristic
148C2×C83D4212yes—
149C2×C83#54612yes—
150D83—42no—
151#2408—831nomaximal, Sylow
152C83#240811yescharacteristic, complemented, Sylow
153C23×D4D8311yescharacteristic, complemented
154C2×#121—832no—
155C2×#120—832no—
156C2×C4×D4—831no—
157#394—832no—
158#392—832no—
159#390—832no—
160#380—831no—
161#379—832no—
162C25C2×D8312yes—
163C22×D4—28no—
164C22×D4—24no—
165C22×D4—834no—
166C22×D4—831no—
167C22×D4C2×D8312yes—
168C22×D4C2×D8312yes—
169C23×C4—831no—
170C23×C4C2×D8311yescharacteristic
171#121—1668no—
172#120—1668no—
173C4×D4—1664no—
174C2×C4⋊C4—831no—
175C2×C4⋊C4—832no—
176C2×C22⋊C4—834no—
177C2×C22⋊C4—831no—
178C2×C22⋊C4—832no—
179C2×C22⋊C4—834no—
180C2×C22⋊C4—834no—
181C2×C22⋊C4—831no—
182C2×C42—831no—
183#95—832no—
184C24—28no—
185C24—24no—
186C24—24no—
187C24—24no—
188C24—44no—
189C24—831no—
190C24C22×D8314yes—
191C24C22×D8311yescharacteristic
192C2×D4—232no—
193C2×D4—28no—
194C2×D4—24no—
195C2×D4—24no—
196C2×D4—24no—
197C2×D4—3324no—
198C2×D4—834no—
199C2×D4—1668no—
200C2×D4—1668no—
201C2×D4—1662no—
202C2×D4—1662no—
203C2×D4C22⋊D8318yescomplemented
204C22×C4—24no—
205C22×C4—831no—
206C22×C4—834no—
207C22×C4—831no—
208C22×C4—831no—
209C22×C4—1662no—
210C22×C4—1664no—
211C22×C4C22⋊D8314yescomplemented
212C22×C4C22×D8311yescharacteristic
213C22×C4C22×D8311yescharacteristic
214C4⋊C4—1662no—
215C4⋊C4—1664no—
216C22⋊C4—3324no—
217C22⋊C4—1668no—
218C22⋊C4—1664no—
219C22⋊C4—1662no—
220C22⋊C4—1668no—
221C22⋊C4—1662no—
222C42—1662no—
223C23—22no—
224C23—22no—
225C23—216no—
226C23—24no—
227C23—24no—
228C23—24no—
229C23—24no—
230C23—24no—
231C23—24no—
232C23—416no—
233C23—416no—
234C23—48no—
235C23—48no—
236C23—832no—
237C23—1661no—
238C23—1662no—
239C23—1661no—
240C23—1662no—
241C23D4×D8312yes—
242C23C2×C22⋊D8318yes—
243C23C2×C22⋊D8312yes—
244C23#1305058912yes—
245C23C23×D8311yescharacteristic
246D4—216no—
247D4—216no—
248D4—33216no—
249D4—1668no—
250C2×C4—22no—
251C2×C4—28no—
252C2×C4—3324no—
253C2×C4—832no—
254C2×C4—1664no—
255C2×C4—1661no—
256C2×C4—1662no—
257C2×C4—1661no—
258C2×C4—1664no—
259C2×C4—1662no—
260C2×C4—1664no—
261C2×C4—1661no—
262C2×C4—1661no—
263C2×C4C2×C22⋊D8312yes—
264C2×C4C2×C22⋊D8312yes—
265C2×C4C2×C22⋊D8314yes—
266C22—24no—
267C22—24no—
268C22—22no—
269C22—22no—
270C22—28no—
271C22—28no—
272C22—28no—
273C22—432no—
274C22—44no—
275C22—44no—
276C22—44no—
277C22—44no—
278C22—3322no—
279C22—3321no—
280C22—3321no—
281C22—1664no—
282C22—1662no—
283C22#1458176314yescomplemented
284C22C2×D4×D8311yescharacteristic
285C22C22×C22⋊D8312yes—
286C22C22×C22⋊D8311yescharacteristic
287C22C2×#1305058911yescharacteristic
288C22#1458183712yes—
289C4—22no—
290C4—3324no—
291C4—3321no—
292C4—3321no—
293C4—1662no—
294C4#1458176114yescomplemented
295C2—24no—
296C2—48no—
297C2—3322no—
298C2C2×#1458176111yescharacteristic
299C2C2×#1458176311yescharacteristic
300C2#1912826811yescharacteristic
301C2D4×C22⋊D8312yes—
302C2#1912834512yes—
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
FittingC23×D4×C83C2
hypercenterC23×D4D83
hypocenterC83#2408
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ C23×D4

Complements

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

NH ≅ G/Ntype
1. C2×#14581763C2semidirect
2. C2×#14581761C2semidirect
3. C2×C4×C22⋊D83C2semidirect
4. #19127615C2semidirect
5. #19127609C2semidirect
6. #19127604C2semidirect
7. #19127561C2semidirect
8. #19127557C2semidirect
9. C23×D4×C83C2semidirect
11. C22×C22⋊D83C22semidirect
12. C22×C4×D83C22semidirect
16. C2×C4⋊Dic83C22semidirect
17. C2×Dic83⋊C4C22semidirect
20. C2×C22⋊Dic83C22semidirect
21. C2×C22⋊Dic83C22semidirect
22. C2×C22⋊Dic83C22semidirect
23. C2×(C2×D83)⋊C4C22semidirect
24. C2×C4×Dic83C22semidirect
25. C83⋊#95C22semidirect
38. C2×C22⋊D83D4semidirect
66. C2×D4×C83D4semidirect
68. C22×C4×C83D4semidirect
93. C22×D83C2×D4semidirect
112. C2×Dic83C2×D4semidirect
126. C22×C83#121semidirect
138. C4×C83#120semidirect
152. C83#2408semidirect
153. C23×D4D83semidirect
203. C2×D4C22⋊D83semidirect
211. C22×C4C22⋊D83semidirect
283. C22#14581763semidirect
294. C4#14581761semidirect

Maximal subgroups

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

subgroupindex
C2×#145817632
C2×#145817612
C2×C4×C22⋊D832
#191276152
#191276092
#191276042
#191275612
#191275572
C23×D4×C832
#240883