Finite Group Databasebeta

← #24917513

Subgroups — #24917513

Order 10624.

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

normal subgroups only

This group has 454 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#19128237C211yesmaximal, characteristic, complemented
2C2×#14581754C211yesmaximal, characteristic, complemented
3#19128019C211yesmaximal, characteristic, complemented
4#19128004C212yesmaximal, complemented
5#19127919C212yesmaximal, complemented
6#19127851C211yesmaximal, characteristic, complemented
7#19127837C211yesmaximal, characteristic
8#19127786C212yesmaximal, complemented
9#19127783C212yesmaximal, complemented
10#19127751C211yesmaximal, characteristic, complemented
11#484×C83C211yesmaximal, characteristic, complemented
12C22×D332C2211yescharacteristic
13C22×C4×D83—21no—
14C2×#13050577C2212yes—
15C2×D664C2212yes—
16#14581787—21no—
17#14581783C2211yescharacteristic, complemented
18#14581781C2211yescharacteristic, complemented
19#14581776—21no—
20#14581766C2211yescharacteristic
21Q8×Dic83C2211yescharacteristic, complemented
22#14581754—21no—
23#14581754—21no—
24#14581754C2212yes—
25#14581754C2212yes—
26#14581754C2211yescharacteristic, complemented
27(C22⋊D83)⋊C4—21no—
28#14581750—21no—
29C2×(C2×D83)⋊C4—21no—
30C2×C4×Dic83C2211yescharacteristic
31#14581723C2212yescomplemented
32Dic83⋊C8C2212yescomplemented
33#14581714C2212yes—
34#14581712C2212yes—
35#14581712C2212yescomplemented
36C22⋊(C83⋊C8)C2211yescharacteristic
37C8×Dic83C2212yescomplemented
38C22⋊Q8×C83C2211yescharacteristic
39C2×C4⋊C4×C83C2211yescharacteristic
40#107×C83C2212yescomplemented
41#103×C83C2212yescomplemented
42C22⋊C8×C83C2211yescharacteristic
43C23×D83—21no—
44C2×D332—21no—
45C2×D332—24no—
46C2×D332C2311yescharacteristic
47C2×D332C2312yes—
48C2×C22⋊D83—21no—
49C22×Dic83D411yescharacteristic
50C22×Dic83D411yescharacteristic
51C2×C4×D83—22no—
52C2×C4×D83—22no—
53C2×C4×D83—21no—
54C2×C4×D83—44no—
55#13050577—24no—
56D664—24no—
57C2×C83⋊C8C2312yes—
58C4⋊Dic83—21no—
59C4⋊Dic83—21no—
60C4⋊Dic83C2311yescharacteristic
61Dic83⋊C4—21no—
62(C2×D83)⋊C4—21no—
63(C2×D83)⋊C4—21no—
64(C2×D83)⋊C4—21no—
65(C2×D83)⋊C4—21no—
66(C2×D83)⋊C4—22no—
67(C2×D83)⋊C4—22no—
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—21no—
86C22×D83—42no—
87C22×D83—42no—
88Q8×C83—22no—
89C2×C4×C83—22no—
90C2×C4×C83—21no—
91C2×C4×C83—21no—
92C2×C4×C83—22no—
93C2×C4×C83—42no—
94C2×C4×C83C2×D411yescharacteristic
95C2×C4×C83C2411yescharacteristic
96C22⋊D83—42no—
97C2×Dic83—22no—
98C2×Dic83—22no—
99C2×Dic83—21no—
100C2×Dic83C2×D411yescharacteristic
101C2×Dic83C2×D411yescharacteristic
102C2×Dic83C2×D411yescharacteristic
103C2×Dic83C2×D411yescharacteristic
104D332—24no—
105D332—22no—
106D332—22no—
107D332#4114yes—
108C4×D83—44no—
109C4×D83—42no—
110C4×D83—44no—
111C8×C83—22no—
112C83⋊C8—22no—
113C22×C83—21no—
114C22×C83—21no—
115C22×C83#13512yes—
116C22×C83C22×D411yescharacteristic
117C2×D83—24no—
118C2×D83—48no—
119C2×D83—42no—
120C2×D83—42no—
121C2×D83—41no—
122C2×D83—41no—
123C2×D83—41no—
124C4×C83—21no—
125C4×C83—41no—
126C4×C83—42no—
127C4×C83—41no—
128C4×C83C2×#4112yes—
129Dic83—21no—
130Dic83—21no—
131Dic83—21no—
132Dic83—41no—
133Dic83AGL(1,ℤ/8)12yes—
134C2×C83—22no—
135C2×C83#52511yescharacteristic
136C2×C83C2×#13511yescharacteristic
137C2×C83C2×AGL(1,ℤ/8)11yescharacteristic
138D83—44no—
139D83—81no—
140#3105—831nomaximal, Sylow
141C83#310511yescharacteristic, complemented, Sylow
142#529—831no—
143C2×C4×D4—831no—
144#484D8311yescharacteristic, complemented
145#481—831no—
146#478—832no—
147#463—832no—
148#450—831no—
149#447—831no—
150C4×SD16—832no—
151C4×D8—832no—
152#431—831no—
153C22×D4—831no—
154C23×C4—1661no—
155C2×SD16—832no—
156C2×D8—832no—
157#126—1661no—
158#125—831no—
159#124—831no—
160#123—1661no—
161C22⋊Q8C2×D8311yescharacteristic
162#121—831no—
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⋊C4C2×D8311yescharacteristic
172C2×C22⋊C4—1661no—
173C2×C42—831no—
174#107C2×D8312yescomplemented
175#106—832no—
176C4⋊C8—832no—
177#103C2×D8312yescomplemented
178#102—832no—
179#102—832no—
180#102—832no—
181C22⋊C8—831no—
182C22⋊C8C2×D8311yescharacteristic
183C4×C8—832no—
184C24—1661no—
185C2×Q8C22×D8311yescharacteristic
186C2×D4—832no—
187C2×D4—831no—
188C2×D4—1664no—
189C2×D4—1661no—
190C2×D4—1661no—
191C22×C4—21no—
192C22×C4—3324no—
193C22×C4—831no—
194C22×C4—831no—
195C22×C4—1661no—
196C22×C4—1662no—
197C22×C4—1662no—
198C22×C4C22×D8311yescharacteristic
199SD16—1664no—
200D8—1664no—
201C2×C8—832no—
202C2×C8C22×D8312yes—
203C4⋊C4—21no—
204C4⋊C4—21no—
205C4⋊C4—831no—
206C4⋊C4—1661no—
207C4⋊C4—1661no—
208C4⋊C4—1661no—
209C4⋊C4C22×D8311yescharacteristic
210C4⋊C4C22×D8312yes—
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—3322no—
224C23—3322no—
225C23—1664no—
226C23—1661no—
227C23—1662no—
228C23D4×D8311yescharacteristic
229Q8—22no—
230D4—3322no—
231D4—834no—
232D4—1664no—
233D4—1662no—
234D4—1662no—
235C2×C4—22no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—22no—
239C2×C4—42no—
240C2×C4—3324no—
241C2×C4—3322no—
242C2×C4—3324no—
243C2×C4—831no—
244C2×C4—831no—
245C2×C4—831no—
246C2×C4—831no—
247C2×C4—1661no—
248C2×C4—1662no—
249C2×C4—1662no—
250C2×C4D4×D8311yescharacteristic
251C2×C4C23×D8311yescharacteristic
252C8—22no—
253C8—1662no—
254C22—21no—
255C22—21no—
256C22—3328no—
257C22—3322no—
258C22—3322no—
259C22—3321no—
260C22—3321no—
261C22—3321no—
262C22—1664no—
263C22#1458181512yes—
264C22C2×D4×D8311yescharacteristic
265C4—21no—
266C4—41no—
267C4—42no—
268C4—41no—
269C4—3321no—
270C4—832no—
271C4—1661no—
272C4—1661no—
273C4—1661no—
274C4#41×D8312yes—
275C2—22no—
276C2—3324no—
277C2—6641no—
278C2#1912820911yescharacteristic
279C2C2×#1458181511yescharacteristic
280C2AGL(1,ℤ/8)×D8311yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#484×C83C2
socleC22×C83C22×D4
prime radical#19127837C2
hypercenter#484D83
hypocenterC83#3105
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. #19128237C2semidirect
2. C2×#14581754C2semidirect
3. #19128019C2semidirect
4. #19128004C2semidirect
5. #19127919C2semidirect
6. #19127851C2semidirect
8. #19127786C2semidirect
9. #19127783C2semidirect
10. #19127751C2semidirect
11. #484×C83C2semidirect
17. #14581783C22semidirect
18. #14581781C22semidirect
21. Q8×Dic83C22semidirect
26. #14581754C22semidirect
31. #14581723C22semidirect
32. Dic83⋊C8C22semidirect
35. #14581712C22semidirect
37. C8×Dic83C22semidirect
40. #107×C83C22semidirect
41. #103×C83C22semidirect
141. C83#3105semidirect
144. #484D83semidirect
174. #107C2×D83semidirect
177. #103C2×D83semidirect

Maximal subgroups

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

subgroupindex
#191282372
C2×#145817542
#191280192
#191280042
#191279192
#191278512
#191278372
#191277862
#191277832
#191277512
#484×C832
#310583