Finite Group Databasebeta

← #24917551

Subgroups — #24917551

Order 10624.

488 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 488 conjugacy classes of subgroups — too many to draw. Only its 119 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#19128246C211yesmaximal, characteristic, complemented
2C2×#14581754C211yesmaximal, characteristic, complemented
3#19128019C211yesmaximal, characteristic, complemented
4#19127990C212yesmaximal, complemented
5#19127895C212yesmaximal, complemented
6#19127837C211yesmaximal, characteristic, complemented
7#19127834C211yesmaximal, characteristic, complemented
8#19127812C212yesmaximal, complemented
9#19127810C212yesmaximal, complemented
10#19127755C211yesmaximal, characteristic, complemented
11#481×C83C211yesmaximal, characteristic, complemented
12C2×#13050589—21no—
13C22×D332C2211yescharacteristic
14C22×C4×D83—21no—
15C2×D664C2212yes—
16C2×#13050575C2212yes—
17#14581791C2211yescharacteristic, complemented
18#14581788C2211yescharacteristic, complemented
19#14581778—21no—
20#14581766C2211yescharacteristic
21#14581764—21no—
22D4×Dic83C2211yescharacteristic, complemented
23#14581754—21no—
24#14581754—21no—
25#14581754C2212yes—
26#14581754C2212yes—
27#14581754C2211yescharacteristic, complemented
28(C22⋊D83)⋊C4—21no—
29C2×(C2×D83)⋊C4—21no—
30C2×C4×Dic83C2211yescharacteristic
31#14581723C2212yescomplemented
32Dic83⋊C8C2212yescomplemented
33#14581714C2212yes—
34#14581712C2212yes—
35#14581712C2212yescomplemented
36C22⋊(C83⋊C8)C2211yescharacteristic
37#14581702C2212yescomplemented
38#121×C83C2211yescharacteristic
39C2×C4⋊C4×C83C2211yescharacteristic
40#106×C83C2212yescomplemented
41#102×C83C2212yescomplemented
42C22⋊C8×C83C2211yescharacteristic
43C23×D83—21no—
44#13050589—44no—
45C2×Dic166—21no—
46C2×D332—21no—
47C2×D332—24no—
48C2×D332C2311yescharacteristic
49C2×D332C2312yes—
50C2×C22⋊D83—21no—
51C2×C22⋊D83—21no—
52C2×C22⋊D83—21no—
53C22×Dic83—21no—
54C22×Dic83D411yescharacteristic
55C22×Dic83D411yescharacteristic
56C2×C4×D83—22no—
57C2×C4×D83—22no—
58C2×C4×D83—21no—
59C2×C4×D83—44no—
60D664—24no—
61#13050575—24no—
62C2×C83⋊C8C2312yes—
63C4⋊Dic83C2311yescharacteristic
64Dic83⋊C4—21no—
65C22⋊Dic83—21no—
66(C2×D83)⋊C4—21no—
67(C2×D83)⋊C4—21no—
68(C2×D83)⋊C4—22no—
69(C2×D83)⋊C4—22no—
70C4×Dic83D411yescharacteristic
71C4×Dic83D411yescharacteristic
72C4×Dic83C2311yescharacteristic
73C4×Dic83C2311yescharacteristic
74C2×D4×C83—21no—
75C2×D4×C83C2311yescharacteristic
76C22×C4×C83—21no—
77C22×C4×C83C2311yescharacteristic
78C2×C8×C83C2312yes—
79C4⋊C4×C83—21no—
80C4⋊C4×C83C2311yescharacteristic
81C4⋊C4×C83C2312yes—
82C22⋊C4×C83—21no—
83C23×C83—21no—
84C23×C83C2×D411yescharacteristic
85C22×D83—24no—
86C22×D83—22no—
87C22×D83—21no—
88C22×D83—42no—
89C22×D83—42no—
90D4×C83—22no—
91D4×C83—42no—
92C2×C4×C83—22no—
93C2×C4×C83—21no—
94C2×C4×C83—22no—
95C2×C4×C83—42no—
96C2×C4×C83C2×D411yescharacteristic
97C2×C4×C83C2411yescharacteristic
98C22⋊D83—42no—
99C22⋊D83—42no—
100C22⋊D83—42no—
101C2×Dic83—22no—
102C2×Dic83—22no—
103C2×Dic83—21no—
104C2×Dic83—42no—
105C2×Dic83C2×D411yescharacteristic
106C2×Dic83C2×D411yescharacteristic
107C2×Dic83C2×D411yescharacteristic
108C2×Dic83C2×D411yescharacteristic
109D332—24no—
110D332—22no—
111D332—22no—
112D332#4114yescomplemented
113C4×D83—44no—
114C4×D83—42no—
115C4×D83—44no—
116Dic166—42no—
117C8×C83—22no—
118C83⋊C8—22no—
119C22×C83—21no—
120C22×C83—21no—
121C22×C83—41no—
122C22×C83—41no—
123C22×C83—41no—
124C22×C83AGL(1,ℤ/8)12yes—
125C22×C83C22×D411yescharacteristic
126C2×D83—24no—
127C2×D83—48no—
128C2×D83—42no—
129C2×D83—42no—
130C2×D83—41no—
131C2×D83—41no—
132C2×D83—41no—
133C4×C83—21no—
134C4×C83—41no—
135C4×C83—42no—
136C4×C83C2×#4112yes—
137Dic83—21no—
138Dic83—21no—
139Dic83—21no—
140Dic83—41no—
141Dic83AGL(1,ℤ/8)12yes—
142C2×C83—22no—
143C2×C83—81no—
144C2×C83#52511yescharacteristic
145C2×C83C2×AGL(1,ℤ/8)11yescharacteristic
146C2×C83C2×AGL(1,ℤ/8)11yescharacteristic
147D83—44no—
148D83—81no—
149#3109—831nomaximal, Sylow
150C83#310911yescharacteristic, complemented, Sylow
151#532—831no—
152C2×C4×D4—831no—
153#481—831no—
154#481D8311yescharacteristic, complemented
155#476—832no—
156#459—832no—
157#447—831no—
158#447—831no—
159#442—832no—
160#442—832no—
161#432—831no—
162C2×#41—1661no—
163C22×D4—831no—
164C23×C4—1661no—
165C2×D8—832no—
166C2×D8—832no—
167C4⋊Q8—831no—
168#127—831no—
169#124—1661no—
170#121—831no—
171#121—1661no—
172#121C2×D8311yescharacteristic
173C4×D4—831no—
174C4×D4—832no—
175C4×D4—832no—
176C4×D4—831no—
177C4×D4—1661no—
178C4×D4—1661no—
179C4×D4—1661no—
180C2×C4⋊C4C2×D8311yescharacteristic
181C2×C22⋊C4—1661no—
182C2×C42—831no—
183#106—832no—
184#106C2×D8312yescomplemented
185C4⋊C8—832no—
186#102—832no—
187#102—832no—
188#102—832no—
189#102C2×D8312yescomplemented
190C22⋊C8—831no—
191C22⋊C8C2×D8311yescharacteristic
192#97—832no—
193C24—1661no—
194#41—3324no—
195C2×Q8—1661no—
196C2×D4—21no—
197C2×D4—832no—
198C2×D4—831no—
199C2×D4—1664no—
200C2×D4—1661no—
201C2×D4—1661no—
202C2×D4—1661no—
203C2×D4—1661no—
204C2×D4C22×D8311yescharacteristic
205C22×C4—21no—
206C22×C4—3324no—
207C22×C4—831no—
208C22×C4—831no—
209C22×C4—1662no—
210C22×C4—1662no—
211C22×C4—1661no—
212C22×C4—1661no—
213C22×C4C22×D8311yescharacteristic
214D8—1664no—
215D8—1664no—
216C2×C8—832no—
217C2×C8C22×D8312yes—
218C4⋊C4—21no—
219C4⋊C4—831no—
220C4⋊C4—1661no—
221C4⋊C4C22×D8311yescharacteristic
222C4⋊C4C22×D8312yes—
223C22⋊C4—21no—
224C22⋊C4—1662no—
225C22⋊C4—1662no—
226C22⋊C4—1661no—
227C22⋊C4—1661no—
228C22⋊C4—1661no—
229C42—831no—
230C42—831no—
231C42—831no—
232C42—831no—
233C23—21no—
234C23—3322no—
235C23—3322no—
236C23—1664no—
237C23—1661no—
238C23—1662no—
239C23D4×D8311yescharacteristic
240Q8—3322no—
241D4—22no—
242D4—42no—
243D4—3322no—
244D4—3322no—
245D4—3322no—
246D4—834no—
247D4—1664no—
248D4—1662no—
249D4—1662no—
250C2×C4—22no—
251C2×C4—21no—
252C2×C4—22no—
253C2×C4—42no—
254C2×C4—3324no—
255C2×C4—3324no—
256C2×C4—3322no—
257C2×C4—3322no—
258C2×C4—831no—
259C2×C4—831no—
260C2×C4—831no—
261C2×C4—831no—
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—3328no—
275C22—3322no—
276C22—3322no—
277C22—3321no—
278C22—3321no—
279C22—3321no—
280C22—1664no—
281C22#1458181912yes—
282C22C2×D4×D8311yescharacteristic
283C4—21no—
284C4—42no—
285C4—41no—
286C4—3321no—
287C4—832no—
288C4—1661no—
289C4—1661no—
290C4—1661no—
291C4#41×D8312yes—
292C2—22no—
293C2—81no—
294C2—3324no—
295C2—6641no—
296C2#1912820911yescharacteristic
297C2AGL(1,ℤ/8)×D8311yescharacteristic
298C2C2×#1458181911yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#481×C83C2
socleC22×C83C22×D4
hypercenter#481D83
hypocenterC83#3109
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ #481

Complements

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

NH ≅ G/Ntype
1. #19128246C2semidirect
2. C2×#14581754C2semidirect
3. #19128019C2semidirect
4. #19127990C2semidirect
5. #19127895C2semidirect
6. #19127837C2semidirect
7. #19127834C2semidirect
8. #19127812C2semidirect
9. #19127810C2semidirect
10. #19127755C2semidirect
11. #481×C83C2semidirect
17. #14581791C22semidirect
18. #14581788C22semidirect
22. D4×Dic83C22semidirect
27. #14581754C22semidirect
31. #14581723C22semidirect
32. Dic83⋊C8C22semidirect
35. #14581712C22semidirect
37. #14581702C22semidirect
40. #106×C83C22semidirect
41. #102×C83C22semidirect
112. D332#41semidirect
150. C83#3109semidirect
154. #481D83semidirect
184. #106C2×D83semidirect
189. #102C2×D83semidirect

Maximal subgroups

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

subgroupindex
#191282462
C2×#145817542
#191280192
#191279902
#191278952
#191278372
#191278342
#191278122
#191278102
#191277552
#481×C832
#310983