Finite Group Databasebeta

← #24917466

Subgroups — #24917466

Order 10624.

448 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 448 conjugacy classes of subgroups — too many to draw. Only its 129 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#19128253C211yesmaximal, characteristic, complemented
2C2×D4×Dic83C211yesmaximal, characteristic
3#19128032C211yesmaximal, characteristic, complemented
4#19128029C211yesmaximal, characteristic, complemented
5#19127989C212yesmaximal, complemented
6#19127904C212yesmaximal, complemented
7#19127852C211yesmaximal, characteristic
8#19127792C212yesmaximal, complemented
9SD16×Dic83C212yesmaximal, complemented
10#19127751C211yesmaximal, characteristic, complemented
11#450×C83C211yesmaximal, characteristic
12C23×Dic83—21no—
13C2×#13050578C2212yes—
14#14581791C2211yescharacteristic, complemented
15#14581790C2211yescharacteristic, complemented
16#14581782—21no—
17#14581771C2211yescharacteristic
18#14581769—21no—
19Q8×Dic83C2211yescharacteristic, complemented
20#14581757—21no—
21#14581757C2211yescharacteristic, complemented
22D4×Dic83—21no—
23D4×Dic83—21no—
24D4×Dic83C2212yes—
25D4×Dic83C2212yes—
26#14581748—21no—
27C2×C4⋊Dic83C2211yescharacteristic
28C2×C22⋊Dic83—21no—
29C2×C4×Dic83C2211yescharacteristic
30#14581724C2212yescomplemented
31#14581723C2212yescomplemented
32Dic83⋊C8C2212yescomplemented
33#14581717C2212yescomplemented
34#14581716C2212yescomplemented
35#14581713C2212yes—
36C22⋊(C83⋊C8)C2211yescharacteristic
37C8×Dic83C2212yescomplemented
38C22×D4×C83C2211yescharacteristic
39C2×SD16×C83C2212yes—
40C22⋊Q8×C83C2211yescharacteristic
41#102×C83C2212yes—
42C22⋊C8×C83C2211yescharacteristic
43C2×Dic166—21no—
44C2×Dic166C2311yescharacteristic
45C22×Dic83—21no—
46C22×Dic83—22no—
47C22×Dic83—22no—
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—
60Dic83⋊C4—21no—
61Dic83⋊C4—21no—
62C22⋊Dic83—21no—
63C22⋊Dic83—22no—
64C22⋊Dic83—22no—
65C4×Dic83—21no—
66C4×Dic83—21no—
67C4×Dic83D411yescharacteristic
68C4×Dic83D411yescharacteristic
69C4×Dic83C2311yescharacteristic
70C4×Dic83C2311yescharacteristic
71C24×C83—21no—
72C2×Q8×C83C2311yescharacteristic
73C2×D4×C83—24no—
74C2×D4×C83—21no—
75C2×D4×C83C2312yes—
76C22×C4×C83C2311yescharacteristic
77SD16×C83—24no—
78C2×C8×C83C2312yes—
79C4⋊C4×C83—21no—
80C4⋊C4×C83C2311yescharacteristic
81C22⋊C4×C83—21no—
82C23×C83—24no—
83C23×C83—22no—
84C23×C83—42no—
85C23×C83—42no—
86C23×C83C2×D411yescharacteristic
87Q8×C83—22no—
88D4×C83—24no—
89D4×C83—22no—
90D4×C83#4114yes—
91C2×C4×C83—22no—
92C2×C4×C83—21no—
93C2×C4×C83—21no—
94C2×C4×C83C2×D411yescharacteristic
95C2×C4×C83C2411yescharacteristic
96C2×Dic83—21no—
97C2×Dic83—21no—
98C2×Dic83—22no—
99C2×Dic83—41no—
100C2×Dic83—42no—
101C2×Dic83—44no—
102C2×Dic83—44no—
103C2×Dic83SD1614yescomplemented
104C2×Dic83C2×D411yescharacteristic
105C2×Dic83C2×D411yescharacteristic
106C2×Dic83C2×D411yescharacteristic
107C2×Dic83C2×D411yescharacteristic
108Dic166—22no—
109Dic166—42no—
110C8×C83—22no—
111C83⋊C8—22no—
112C22×C83—21no—
113C22×C83—21no—
114C22×C83—24no—
115C22×C83—48no—
116C22×C83—42no—
117C22×C83—42no—
118C22×C83C2×SD1612yes—
119C22×C83C22×D411yescharacteristic
120C4×C83—21no—
121C4×C83—41no—
122C4×C83—41no—
123C4×C83C2×#4112yes—
124Dic83—21no—
125Dic83—21no—
126Dic83—41no—
127Dic83—42no—
128Dic83—41no—
129Dic83C2×SD1612yes—
130Dic83C2×SD1612yes—
131C2×C83—22no—
132C2×C83—44no—
133C2×C83#52511yescharacteristic
134C2×C83C22×SD1611yescharacteristic
135C2×C83#57511yescharacteristic
136#3101—831nomaximal, Sylow
137C83#310111yescharacteristic, complemented, Sylow
138#533—831no—
139C2×C4×D4—831no—
140#483—831no—
141#483—831no—
142#476—832no—
143#461—832no—
144#450—831no—
145#450D8311yescharacteristic
146C4×SD16—832no—
147C4×SD16—832no—
148#431—831no—
149C22×D4C2×D8311yescharacteristic
150C23×C4—1661no—
151C2×SD16—832no—
152C2×SD16C2×D8312yes—
153C4⋊Q8—831no—
154C4⋊Q8—831no—
155#125—1661no—
156C22⋊Q8—831no—
157C22⋊Q8—1661no—
158C22⋊Q8C2×D8311yescharacteristic
159C4×Q8—831no—
160C4×Q8—831no—
161C4×Q8—1661no—
162C4×D4—832no—
163C4×D4—832no—
164C4×D4—1661no—
165C4×D4—1661no—
166#117—1661no—
167C2×C4⋊C4—831no—
168C2×C22⋊C4—1661no—
169C2×C42—831no—
170#106—832no—
171#106—832no—
172C4⋊C8—832no—
173#103—832no—
174#103—832no—
175#102—832no—
176#102C2×D8312yes—
177C22⋊C8—831no—
178C22⋊C8C2×D8311yescharacteristic
179C4×C8—832no—
180C24—21no—
181C2×Q8—831no—
182C2×Q8—1661no—
183C2×Q8C22×D8311yescharacteristic
184C2×D4—24no—
185C2×D4—21no—
186C2×D4C22×D8312yes—
187C22×C4—3324no—
188C22×C4—831no—
189C22×C4—831no—
190C22×C4—1662no—
191C22×C4—1662no—
192C22×C4—1661no—
193C22×C4C22×D8311yescharacteristic
194SD16—24no—
195SD16—1664no—
196C2×C8—832no—
197C2×C8C22×D8312yes—
198C4⋊C4—21no—
199C4⋊C4—831no—
200C4⋊C4—832no—
201C4⋊C4—1661no—
202C4⋊C4—1661no—
203C4⋊C4—1661no—
204C4⋊C4—1661no—
205C4⋊C4—1661no—
206C4⋊C4—1661no—
207C4⋊C4—1661no—
208C4⋊C4C22×D8311yescharacteristic
209C22⋊C4—21no—
210C22⋊C4—1662no—
211C22⋊C4—1662no—
212C22⋊C4—1661no—
213C42—831no—
214C42—831no—
215C42—831no—
216C42—831no—
217C42—1661no—
218C42—1661no—
219C23—24no—
220C23—22no—
221C23—42no—
222C23—42no—
223C23D4×D8311yescharacteristic
224Q8—22no—
225Q8—3322no—
226Q8—1662no—
227D4—24no—
228D4—22no—
229D4#1305058914yes—
230C2×C4—22no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—3322no—
234C2×C4—3324no—
235C2×C4—3324no—
236C2×C4—3321no—
237C2×C4—834no—
238C2×C4—831no—
239C2×C4—831no—
240C2×C4—831no—
241C2×C4—831no—
242C2×C4—1661no—
243C2×C4—1661no—
244C2×C4—1662no—
245C2×C4D4×D8311yescharacteristic
246C2×C4C23×D8311yescharacteristic
247C8—22no—
248C8—1662no—
249C22—21no—
250C22—21no—
251C22—24no—
252C22—48no—
253C22—42no—
254C22—42no—
255C22SD16×D8312yes—
256C22C2×D4×D8311yescharacteristic
257C4—21no—
258C4—41no—
259C4—41no—
260C4—3321no—
261C4—3322no—
262C4—3321no—
263C4—832no—
264C4—832no—
265C4—1661no—
266C4—1661no—
267C4C2×#1305058912yes—
268C2—22no—
269C2—44no—
270C2#1912820711yescharacteristic
271C2C2×SD16×D8311yescharacteristic
272C2#1912849711yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#450×C83C2
socleC22×C83C22×D4
prime radicalC22×D4×C83C22
hypercenter#450D83
hypocenterC83#3101
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. #19128253C2semidirect
3. #19128032C2semidirect
4. #19128029C2semidirect
5. #19127989C2semidirect
6. #19127904C2semidirect
8. #19127792C2semidirect
9. SD16×Dic83C2semidirect
10. #19127751C2semidirect
14. #14581791C22semidirect
15. #14581790C22semidirect
19. Q8×Dic83C22semidirect
21. #14581757C22semidirect
30. #14581724C22semidirect
31. #14581723C22semidirect
32. Dic83⋊C8C22semidirect
33. #14581717C22semidirect
34. #14581716C22semidirect
37. C8×Dic83C22semidirect
103. C2×Dic83SD16semidirect
137. C83#3101semidirect

Maximal subgroups

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

subgroupindex
#191282532
C2×D4×Dic832
#191280322
#191280292
#191279892
#191279042
#191278522
#191277922
SD16×Dic832
#191277512
#450×C832
#310183