Finite Group Databasebeta

← #24917559

Subgroups — #24917559

Order 10624.

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

normal subgroups only

This group has 444 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#19128253C211yesmaximal, characteristic, complemented
2C2×#14581754C211yesmaximal, characteristic
3#19128036C211yesmaximal, characteristic, complemented
4#19127980C212yesmaximal, complemented
5#19127905C212yesmaximal, complemented
6#19127854C211yesmaximal, characteristic
7#19127851C211yesmaximal, characteristic
8#19127804C212yesmaximal, complemented
9#19127800C212yesmaximal, complemented
10#19127755C211yesmaximal, characteristic, complemented
11#484×C83C211yesmaximal, characteristic, complemented
12C22×D332C2211yescharacteristic
13C22×C4×D83—21no—
14C2×#13050579C2212yes—
15C2×#13050577C2212yes—
16#14581791C2211yescharacteristic, complemented
17#14581790C2211yescharacteristic, complemented
18#14581782—21no—
19#14581771C2211yescharacteristic
20#14581769—21no—
21Q8×Dic83C2211yescharacteristic, complemented
22#14581757—21no—
23#14581757C2211yescharacteristic, complemented
24#14581754—21no—
25#14581754—21no—
26#14581754C2212yes—
27#14581754C2212yes—
28#14581748—21no—
29C2×(C2×D83)⋊C4—21no—
30C2×C4×Dic83C2211yescharacteristic
31#14581725C2212yescomplemented
32Dic83⋊C8C2212yescomplemented
33#14581715C2212yescomplemented
34#14581714C2212yes—
35#14581712C2212yes—
36C22⋊(C83⋊C8)C2211yescharacteristic
37#14581702C2212yescomplemented
38C22⋊Q8×C83C2211yescharacteristic
39C2×C4⋊C4×C83C2211yescharacteristic
40#107×C83C2212yescomplemented
41#103×C83C2212yescomplemented
42C22⋊C8×C83C2211yescharacteristic
43C23×D83—21no—
44C2×Dic166—21no—
45C2×Dic166C2311yescharacteristic
46C2×D332—21no—
47C2×D332—24no—
48C2×D332C2312yes—
49C22×Dic83D411yescharacteristic
50C22×Dic83D411yescharacteristic
51C2×C4×D83—22no—
52C2×C4×D83—22no—
53C2×C4×D83—44no—
54#13050579—24no—
55#13050577—24no—
56C2×C83⋊C8C2312yes—
57C4⋊Dic83—21no—
58C4⋊Dic83C2311yescharacteristic
59Dic83⋊C4—21no—
60Dic83⋊C4—21no—
61Dic83⋊C4—21no—
62Dic83⋊C4—21no—
63Dic83⋊C4—21no—
64C22⋊Dic83—21no—
65(C2×D83)⋊C4—22no—
66(C2×D83)⋊C4—22no—
67C4×Dic83—21no—
68C4×Dic83—21no—
69C4×Dic83D411yescharacteristic
70C4×Dic83D411yescharacteristic
71C4×Dic83C2311yescharacteristic
72C4×Dic83C2311yescharacteristic
73C2×Q8×C83C2311yescharacteristic
74C22×C4×C83—21no—
75C22×C4×C83C2311yescharacteristic
76C2×C8×C83C2312yes—
77C4⋊C4×C83—21no—
78C4⋊C4×C83—21no—
79C4⋊C4×C83C2311yescharacteristic
80C4⋊C4×C83C2312yes—
81C22⋊C4×C83—21no—
82C23×C83C2×D411yescharacteristic
83C22×D83—24no—
84C22×D83—22no—
85C22×D83—42no—
86C22×D83—42no—
87Q8×C83—22no—
88C2×C4×C83—22no—
89C2×C4×C83—21no—
90C2×C4×C83—21no—
91C2×C4×C83—22no—
92C2×C4×C83—42no—
93C2×C4×C83C2×D411yescharacteristic
94C2×C4×C83C2411yescharacteristic
95C2×Dic83—22no—
96C2×Dic83—22no—
97C2×Dic83—21no—
98C2×Dic83—21no—
99C2×Dic83C2×D411yescharacteristic
100C2×Dic83C2×D411yescharacteristic
101C2×Dic83C2×D411yescharacteristic
102C2×Dic83C2×D411yescharacteristic
103D332—24no—
104D332—22no—
105D332#4114yes—
106C4×D83—44no—
107C4×D83—44no—
108Dic166—22no—
109Dic166—42no—
110C8×C83—22no—
111C83⋊C8—22no—
112C22×C83—21no—
113C22×C83—21no—
114C22×C83#13712yes—
115C22×C83C22×D411yescharacteristic
116C2×D83—24no—
117C2×D83—48no—
118C2×D83—42no—
119C2×D83—42no—
120C4×C83—21no—
121C4×C83—41no—
122C4×C83—41no—
123C4×C83—42no—
124C4×C83C2×#4112yes—
125Dic83—21no—
126Dic83—21no—
127Dic83—21no—
128Dic83—41no—
129Dic83—41no—
130Dic83AGL(1,ℤ/8)12yes—
131C2×C83—22no—
132C2×C83#52511yescharacteristic
133C2×C83C2×AGL(1,ℤ/8)11yescharacteristic
134C2×C83C2×#13711yescharacteristic
135D83—44no—
136#3110—831nomaximal, Sylow
137C83#311011yescharacteristic, complemented, Sylow
138#533—831no—
139C2×C4×D4—831no—
140#484—831no—
141#484D8311yescharacteristic, complemented
142#474—832no—
143#461—832no—
144#450—831no—
145#450—831no—
146#440—832no—
147#440—832no—
148#432—831no—
149C22×D4—831no—
150C23×C4—1661no—
151C2×SD16—832no—
152C2×SD16—832no—
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⋊C4C2×D8311yescharacteristic
168C2×C22⋊C4—1661no—
169C2×C42—831no—
170#107—832no—
171#107C2×D8312yescomplemented
172C4⋊C8—832no—
173#103—832no—
174#103C2×D8312yescomplemented
175#102—832no—
176#102—832no—
177C22⋊C8—831no—
178C22⋊C8C2×D8311yescharacteristic
179#97—832no—
180C24—1661no—
181C2×Q8—831no—
182C2×Q8—1661no—
183C2×Q8C22×D8311yescharacteristic
184C2×D4—832no—
185C2×D4—1664no—
186C2×D4—1661no—
187C22×C4—21no—
188C22×C4—3324no—
189C22×C4—831no—
190C22×C4—831no—
191C22×C4—1662no—
192C22×C4—1662no—
193C22×C4C22×D8311yescharacteristic
194SD16—1664no—
195SD16—1664no—
196C2×C8—832no—
197C2×C8C22×D8312yes—
198C4⋊C4—21no—
199C4⋊C4—21no—
200C4⋊C4—831no—
201C4⋊C4—1661no—
202C4⋊C4—1661no—
203C4⋊C4—1661no—
204C4⋊C4—1661no—
205C4⋊C4—1661no—
206C4⋊C4—1661no—
207C4⋊C4C22×D8311yescharacteristic
208C4⋊C4C22×D8312yes—
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—3322no—
220C23—3322no—
221C23—1664no—
222C23—1662no—
223C23D4×D8311yescharacteristic
224Q8—22no—
225Q8—3322no—
226Q8—1662no—
227D4—834no—
228D4—1664no—
229D4—1662no—
230C2×C4—22no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—22no—
234C2×C4—42no—
235C2×C4—3324no—
236C2×C4—3324no—
237C2×C4—831no—
238C2×C4—831no—
239C2×C4—831no—
240C2×C4—831no—
241C2×C4—1661no—
242C2×C4—1661no—
243C2×C4—1662no—
244C2×C4—1662no—
245C2×C4D4×D8311yescharacteristic
246C2×C4C23×D8311yescharacteristic
247C8—22no—
248C8—1662no—
249C22—21no—
250C22—21no—
251C22—3328no—
252C22—3322no—
253C22—3322no—
254C22—1664no—
255C22#1458182412yes—
256C22C2×D4×D8311yescharacteristic
257C4—21no—
258C4—41no—
259C4—42no—
260C4—41no—
261C4—3321no—
262C4—3321no—
263C4—832no—
264C4—1661no—
265C4—1661no—
266C4—1661no—
267C4#41×D8312yes—
268C2—22no—
269C2—3324no—
270C2#1912820911yescharacteristic
271C2AGL(1,ℤ/8)×D8311yescharacteristic
272C2C2×#1458182411yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#484×C83C2
socleC22×C83C22×D4
prime radicalC22×D332C22
hypercenter#484D83
hypocenterC83#3110
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ #484

Complements

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

NH ≅ G/Ntype
1. #19128253C2semidirect
3. #19128036C2semidirect
4. #19127980C2semidirect
5. #19127905C2semidirect
8. #19127804C2semidirect
9. #19127800C2semidirect
10. #19127755C2semidirect
11. #484×C83C2semidirect
16. #14581791C22semidirect
17. #14581790C22semidirect
21. Q8×Dic83C22semidirect
23. #14581757C22semidirect
31. #14581725C22semidirect
32. Dic83⋊C8C22semidirect
33. #14581715C22semidirect
37. #14581702C22semidirect
40. #107×C83C22semidirect
41. #103×C83C22semidirect
137. C83#3110semidirect
141. #484D83semidirect
171. #107C2×D83semidirect
174. #103C2×D83semidirect

Maximal subgroups

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

subgroupindex
#191282532
C2×#145817542
#191280362
#191279802
#191279052
#191278542
#191278512
#191278042
#191278002
#191277552
#484×C832
#311083