Finite Group Databasebeta

← #24917596

Subgroups — #24917596

Order 10624.

450 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 450 conjugacy classes of subgroups — too many to draw. Only its 121 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#19128232C211yesmaximal, characteristic, complemented
2C2×D4×Dic83C211yesmaximal, characteristic, complemented
3#19128032C211yesmaximal, characteristic, complemented
4#19128012C211yesmaximal, characteristic, complemented
5#19128003C212yesmaximal, complemented
6#19127914C212yesmaximal, complemented
7#19127852C211yesmaximal, characteristic, complemented
8#19127813C212yesmaximal, complemented
9#19127798C212yesmaximal, complemented
10#19127755C211yesmaximal, characteristic, complemented
11#447×C83C211yesmaximal, characteristic
12C23×Dic83—21no—
13C2×#13050578C2212yes—
14#14581785—21no—
15#14581783C2211yescharacteristic, complemented
16#14581779C2211yescharacteristic, complemented
17#14581773—21no—
18#14581771C2211yescharacteristic
19#14581757C2211yescharacteristic, complemented
20D4×Dic83—21no—
21D4×Dic83—21no—
22D4×Dic83—21no—
23D4×Dic83C2212yes—
24D4×Dic83C2212yes—
25D4×Dic83C2211yescharacteristic, complemented
26#14581748—21no—
27C2×C4⋊Dic83C2211yescharacteristic
28C2×C22⋊Dic83—21no—
29C2×C4×Dic83C2211yescharacteristic
30#14581725C2212yescomplemented
31#14581724C2212yescomplemented
32Dic83⋊C8C2212yescomplemented
33#14581717C2212yescomplemented
34#14581713C2212yes—
35#14581713C2212yescomplemented
36C22⋊(C83⋊C8)C2211yescharacteristic
37#14581702C2212yescomplemented
38C22×D4×C83C2211yescharacteristic
39C2×D8×C83C2212yes—
40#121×C83C2211yescharacteristic
41#102×C83C2212yes—
42C22⋊C8×C83C2211yescharacteristic
43C2×Dic166C2311yescharacteristic
44C22×Dic83—21no—
45C22×Dic83—22no—
46C22×Dic83—22no—
47C22×Dic83—21no—
48C22×Dic83—44no—
49C22×Dic83D411yescharacteristic
50C22×Dic83D411yescharacteristic
51#13050578—24no—
52C2×C83⋊C8C2312yes—
53C4⋊Dic83—21no—
54C4⋊Dic83—21no—
55C4⋊Dic83C2311yescharacteristic
56C4⋊Dic83C2312yes—
57Dic83⋊C4—21no—
58Dic83⋊C4—21no—
59Dic83⋊C4—21no—
60C22⋊Dic83—21no—
61C22⋊Dic83—21no—
62C22⋊Dic83—21no—
63C22⋊Dic83—21no—
64C22⋊Dic83—22no—
65C22⋊Dic83—22no—
66C4×Dic83—21no—
67C4×Dic83D411yescharacteristic
68C4×Dic83D411yescharacteristic
69C4×Dic83C2311yescharacteristic
70C4×Dic83C2311yescharacteristic
71C24×C83—21no—
72C2×D4×C83—24no—
73C2×D4×C83—21no—
74C2×D4×C83—21no—
75C2×D4×C83C2311yescharacteristic
76C2×D4×C83C2312yes—
77C22×C4×C83C2311yescharacteristic
78D8×C83—24no—
79C2×C8×C83C2312yes—
80C4⋊C4×C83C2311yescharacteristic
81C22⋊C4×C83—21no—
82C23×C83—24no—
83C23×C83—22no—
84C23×C83—21no—
85C23×C83—42no—
86C23×C83—42no—
87C23×C83C2×D411yescharacteristic
88D4×C83—24no—
89D4×C83—22no—
90D4×C83—22no—
91D4×C83—42no—
92D4×C83#4114yes—
93C2×C4×C83—22no—
94C2×C4×C83—21no—
95C2×C4×C83C2×D411yescharacteristic
96C2×C4×C83C2411yescharacteristic
97C2×Dic83—22no—
98C2×Dic83—21no—
99C2×Dic83—21no—
100C2×Dic83—22no—
101C2×Dic83—41no—
102C2×Dic83—42no—
103C2×Dic83—44no—
104C2×Dic83—44no—
105C2×Dic83—42no—
106C2×Dic83C2×D411yescharacteristic
107C2×Dic83C2×D411yescharacteristic
108C2×Dic83C2×D411yescharacteristic
109C2×Dic83C2×D411yescharacteristic
110Dic166—22no—
111C8×C83—22no—
112C83⋊C8—22no—
113C22×C83—21no—
114C22×C83—21no—
115C22×C83—24no—
116C22×C83—48no—
117C22×C83—42no—
118C22×C83—42no—
119C22×C83—41no—
120C22×C83—41no—
121C22×C83—41no—
122C22×C83AGL(1,ℤ/8)12yes—
123C22×C83C22×D411yescharacteristic
124C4×C83—21no—
125C4×C83—41no—
126C4×C83C2×#4112yes—
127Dic83—21no—
128Dic83—21no—
129Dic83—21no—
130Dic83—21no—
131Dic83—41no—
132Dic83—42no—
133Dic83—41no—
134C2×C83—22no—
135C2×C83—44no—
136C2×C83—81no—
137C2×C83#52511yescharacteristic
138C2×C83C2×AGL(1,ℤ/8)11yescharacteristic
139C2×C83#57511yescharacteristic
140#3114—831nomaximal, Sylow
141C83#311411yescharacteristic, complemented, Sylow
142#529—831no—
143C2×C4×D4—831no—
144#483—831no—
145#480—831no—
146#478—832no—
147#463—832no—
148#450—831no—
149#447D8311yescharacteristic
150#442—832no—
151#440—832no—
152#432—831no—
153C22×D4C2×D8311yescharacteristic
154C23×C4—1661no—
155C2×SD16—832no—
156C2×D8C2×D8312yes—
157#126—1661no—
158#125—831no—
159#124—831no—
160#123—1661no—
161C22⋊Q8—831no—
162#121C2×D8311yescharacteristic
163C4×Q8—831no—
164C4×D4—832no—
165C4×D4—832no—
166C4×D4—831no—
167C4×D4—1661no—
168C4×D4—1661no—
169C4×D4—1661no—
170#117—1661no—
171C2×C4⋊C4—831no—
172C2×C22⋊C4—1661no—
173C2×C42—831no—
174#107—832no—
175#106—832no—
176C4⋊C8—832no—
177#103—832no—
178#102—832no—
179#102—832no—
180#102C2×D8312yes—
181C22⋊C8—831no—
182C22⋊C8C2×D8311yescharacteristic
183#97—832no—
184C24—21no—
185C2×Q8—831no—
186C2×D4—24no—
187C2×D4—21no—
188C2×D4—21no—
189C2×D4C22×D8311yescharacteristic
190C2×D4C22×D8312yes—
191C22×C4—3324no—
192C22×C4—831no—
193C22×C4—831no—
194C22×C4—1661no—
195C22×C4—1662no—
196C22×C4—1662no—
197C22×C4—1661no—
198C22×C4C22×D8311yescharacteristic
199SD16—1664no—
200D8—24no—
201C2×C8—832no—
202C2×C8C22×D8312yes—
203C4⋊C4—831no—
204C4⋊C4—832no—
205C4⋊C4—1661no—
206C4⋊C4—1661no—
207C4⋊C4—1661no—
208C4⋊C4—1661no—
209C4⋊C4—1661no—
210C4⋊C4C22×D8311yescharacteristic
211C22⋊C4—21no—
212C22⋊C4—1662no—
213C22⋊C4—1662no—
214C22⋊C4—1661no—
215C22⋊C4—1661no—
216C22⋊C4—1661no—
217C22⋊C4—1661no—
218C42—831no—
219C42—831no—
220C42—831no—
221C42—831no—
222C42—1661no—
223C23—24no—
224C23—22no—
225C23—21no—
226C23—42no—
227C23—42no—
228C23D4×D8311yescharacteristic
229Q8—1662no—
230D4—24no—
231D4—22no—
232D4—22no—
233D4—42no—
234D4#1305058914yes—
235C2×C4—22no—
236C2×C4—21no—
237C2×C4—3322no—
238C2×C4—3324no—
239C2×C4—3324no—
240C2×C4—3322no—
241C2×C4—3321no—
242C2×C4—831no—
243C2×C4—831no—
244C2×C4—831no—
245C2×C4—831no—
246C2×C4—1661no—
247C2×C4—1661no—
248C2×C4—1662no—
249C2×C4—1662no—
250C2×C4D4×D8311yescharacteristic
251C2×C4C23×D8311yescharacteristic
252C8—22no—
253C8—1662no—
254C22—21no—
255C22—21no—
256C22—24no—
257C22—48no—
258C22—42no—
259C22—42no—
260C22—41no—
261C22—41no—
262C22—41no—
263C22#1458181612yes—
264C22C2×D4×D8311yescharacteristic
265C4—21no—
266C4—41no—
267C4—3321no—
268C4—3322no—
269C4—3321no—
270C4—1661no—
271C4—1661no—
272C4—1661no—
273C4—1661no—
274C4C2×#1305058912yes—
275C2—22no—
276C2—44no—
277C2—81no—
278C2#1912820711yescharacteristic
279C2C2×#1458181611yescharacteristic
280C2#1912849711yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting, prime radical#447×C83C2
socleC22×C83C22×D4
hypercenter#447D83
hypocenterC83#3114
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ #447

Complements

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

NH ≅ G/Ntype
1. #19128232C2semidirect
2. C2×D4×Dic83C2semidirect
3. #19128032C2semidirect
4. #19128012C2semidirect
5. #19128003C2semidirect
6. #19127914C2semidirect
7. #19127852C2semidirect
8. #19127813C2semidirect
9. #19127798C2semidirect
10. #19127755C2semidirect
15. #14581783C22semidirect
16. #14581779C22semidirect
19. #14581757C22semidirect
25. D4×Dic83C22semidirect
30. #14581725C22semidirect
31. #14581724C22semidirect
32. Dic83⋊C8C22semidirect
33. #14581717C22semidirect
35. #14581713C22semidirect
37. #14581702C22semidirect
141. C83#3114semidirect

Maximal subgroups

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

subgroupindex
#191282322
C2×D4×Dic832
#191280322
#191280122
#191280032
#191279142
#191278522
#191278132
#191277982
#191277552
#447×C832
#311483