Finite Group Databasebeta

← #483×D83 #24917337

Subgroups — #483×D83 #24917337

Order 10624.

460 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 135 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
1C22⋊Q8×D83C211yesmaximal, characteristic
2C2×C4⋊C4×D83C211yesmaximal, characteristic
3#19128033C212yesmaximal, complemented
4#19128032C211yesmaximal, characteristic, complemented
5#19128031C211yesmaximal, characteristic, complemented
6#19128030C212yesmaximal, complemented
7#19128029C211yesmaximal, characteristic, complemented
8#106×D83C212yesmaximal, complemented
9#103×D83C212yesmaximal, complemented
10C22⋊C8×D83C211yesmaximal, characteristic
11#483×C83C211yesmaximal, characteristic, complemented
12C2×Q8×D83C2211yescharacteristic
13C22×C4×D83—21no—
14C22×C4×D83C2211yescharacteristic
15C2×C8×D83C2212yes—
16#14581772C2211yescharacteristic
17#14581771C2211yescharacteristic
18#14581770—21no—
19#14581769—21no—
20#14581768C2211yescharacteristic
21C2×C4⋊Dic83C2211yescharacteristic
22C2×Dic83⋊C4—21no—
23C4⋊C4×D83—21no—
24C4⋊C4×D83—21no—
25C4⋊C4×D83—21no—
26C4⋊C4×D83C2212yes—
27C4⋊C4×D83C2212yes—
28C4⋊C4×D83C2211yescharacteristic
29C22⋊C4×D83—21no—
30#14581724C2212yescomplemented
31#14581723C2212yescomplemented
32#14581723C2212yescomplemented
33#14581717C2212yescomplemented
34#14581716C2212yescomplemented
35#14581715C2212yescomplemented
36C22⋊(C83⋊C8)C2211yescharacteristic
37(C2×D83)⋊C8C2212yes—
38C22⋊Q8×C83C2211yescharacteristic
39C2×C4⋊C4×C83C2211yescharacteristic
40#106×C83C2212yescomplemented
41#103×C83C2212yescomplemented
42C22⋊C8×C83C2211yescharacteristic
43C23×D83D411yescharacteristic
44C2×Dic166—21no—
45C2×Dic166C2311yescharacteristic
46Q8×D83—24no—
47C22×Dic83—21no—
48C22×Dic83D411yescharacteristic
49C2×C4×D83—24no—
50C2×C4×D83—22no—
51C2×C4×D83—22no—
52C2×C4×D83—21no—
53C2×C4×D83—21no—
54C2×C4×D83—44no—
55C2×C4×D83D411yescharacteristic
56C2×C4×D83D411yescharacteristic
57C2×C4×D83C2311yescharacteristic
58C2×C4×D83C2311yescharacteristic
59C2×C83⋊C8C2312yes—
60C8×D83—24no—
61C4⋊Dic83—21no—
62C4⋊Dic83—21no—
63C4⋊Dic83C2311yescharacteristic
64C4⋊Dic83C2312yes—
65Dic83⋊C4—21no—
66Dic83⋊C4—21no—
67Dic83⋊C4—21no—
68Dic83⋊C4—22no—
69Dic83⋊C4—22no—
70C22⋊Dic83—21no—
71(C2×D83)⋊C4—21no—
72(C2×D83)⋊C4—21no—
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—22no—
84C22×D83—22no—
85C22×D83SD1614yes—
86C22×D83C2×D411yescharacteristic
87C22×D83C2×D411yescharacteristic
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
96C2×Dic83—22no—
97C2×Dic83—21no—
98C2×Dic83—21no—
99C2×Dic83—22no—
100C2×Dic83—42no—
101C2×Dic83C2×D411yescharacteristic
102C2×Dic83C2×D411yescharacteristic
103C4×D83—22no—
104C4×D83—22no—
105C4×D83—22no—
106C4×D83—42no—
107C4×D83—44no—
108C4×D83—44no—
109C4×D83—42no—
110C4×D83#4114yes—
111Dic166—22no—
112Dic166—42no—
113C8×C83—22no—
114C83⋊C8—22no—
115C22×C83—21no—
116C22×C83—21no—
117C22×C83C2×SD1612yes—
118C22×C83C22×D411yescharacteristic
119C2×D83—21no—
120C2×D83—21no—
121C2×D83—28no—
122C2×D83#12312yes—
123C2×D83C2×SD1612yes—
124C2×D83C2×SD1612yes—
125C2×D83#13712yes—
126C4×C83—21no—
127C4×C83—41no—
128C4×C83—42no—
129C4×C83—41no—
130C4×C83C2×#4112yes—
131Dic83—21no—
132Dic83—41no—
133Dic83—42no—
134Dic83—41no—
135Dic83#12312yes—
136C2×C83—22no—
137C2×C83C2×#12311yescharacteristic
138C2×C83C22×SD1611yescharacteristic
139C2×C83C2×#13711yescharacteristic
140D83—22no—
141D83#48314yesdirect product factor
142C2×#483—831nomaximal, Sylow
143C83C2×#48311yescharacteristic, complemented, Sylow
144C2×C22⋊Q8—831no—
145C22×C4⋊C4—831no—
146#483—831no—
147#483—831no—
148#483—832no—
149#483—832no—
150#483—831no—
151#483D8311yescharacteristic, direct product factor
152C2×#106—832no—
153C2×#103—832no—
154C2×C22⋊C8—831no—
155C22×Q8—831no—
156C23×C4—831no—
157C23×C4—1661no—
158C22×C8—832no—
159C22⋊Q8—831no—
160C22⋊Q8—831no—
161C22⋊Q8—831no—
162C22⋊Q8—1661no—
163C22⋊Q8—1661no—
164C22⋊Q8C2×D8311yescharacteristic
165C2×C4⋊C4—831no—
166C2×C4⋊C4—832no—
167C2×C4⋊C4—832no—
168C2×C4⋊C4—831no—
169C2×C4⋊C4—1661no—
170C2×C4⋊C4—1661no—
171C2×C4⋊C4—1661no—
172C2×C4⋊C4—1661no—
173C2×C4⋊C4C2×D8311yescharacteristic
174C2×C22⋊C4—1661no—
175#106—832no—
176#106—832no—
177#106—832no—
178#106C2×D8312yescomplemented
179#103—832no—
180#103—832no—
181#103—832no—
182#103C2×D8312yescomplemented
183C22⋊C8—831no—
184C22⋊C8—832no—
185C22⋊C8C2×D8311yescharacteristic
186C24—831no—
187C2×Q8—831no—
188C2×Q8—1661no—
189C2×Q8—1664no—
190C2×Q8C22×D8311yescharacteristic
191C22×C4—21no—
192C22×C4—3324no—
193C22×C4—831no—
194C22×C4—831no—
195C22×C4—831no—
196C22×C4—831no—
197C22×C4—831no—
198C22×C4—1661no—
199C22×C4—1661no—
200C22×C4—1662no—
201C22×C4—1662no—
202C22×C4—1664no—
203C22×C4—1661no—
204C22×C4C22×D8311yescharacteristic
205C2×C8—832no—
206C2×C8—1664no—
207C2×C8C22×D8312yes—
208C4⋊C4—21no—
209C4⋊C4—21no—
210C4⋊C4—832no—
211C4⋊C4—831no—
212C4⋊C4—1662no—
213C4⋊C4—1662no—
214C4⋊C4—1661no—
215C4⋊C4—1661no—
216C4⋊C4—1661no—
217C4⋊C4—1661no—
218C4⋊C4—1661no—
219C4⋊C4C22×D8312yes—
220C4⋊C4C22×D8311yescharacteristic
221C22⋊C4—21no—
222C22⋊C4—1661no—
223C22⋊C4—1661no—
224C22⋊C4—1661no—
225C23—831no—
226C23—834no—
227C23—831no—
228C23—1662no—
229C23—1662no—
230C23D4×D8311yescharacteristic
231Q8—22no—
232Q8—3322no—
233Q8—1662no—
234C2×C4—22no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—22no—
238C2×C4—42no—
239C2×C4—3324no—
240C2×C4—3324no—
241C2×C4—3322no—
242C2×C4—3322no—
243C2×C4—3322no—
244C2×C4—831no—
245C2×C4—834no—
246C2×C4—831no—
247C2×C4—1662no—
248C2×C4—1662no—
249C2×C4—1661no—
250C2×C4—1661no—
251C2×C4—1662no—
252C2×C4—1662no—
253C2×C4—1662no—
254C2×C4D4×D8311yescharacteristic
255C2×C4C23×D8311yescharacteristic
256C8—22no—
257C8—1662no—
258C22—21no—
259C22—21no—
260C22—832no—
261C22—832no—
262C22—832no—
263C22—832no—
264C22—1668no—
265C22—1661no—
266C22—1661no—
267C22SD16×D8312yes—
268C22C2×D4×D8311yescharacteristic
269C4—21no—
270C4—41no—
271C4—42no—
272C4—41no—
273C4—3321no—
274C4—3322no—
275C4—3321no—
276C4—832no—
277C4—1661no—
278C4#41×D8312yes—
279C2—22no—
280C2—834no—
281C2—1662no—
282C2#123×D8311yescharacteristic
283C2C2×SD16×D8311yescharacteristic
284C2#137×D8311yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#483×C83C2
socleC22×C83C22×D4
prime radicalC23×D83D4
hypercenter#483D83
hypocenterC83C2×#483
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ #483

Complements

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

NH ≅ G/Ntype
3. #19128033C2semidirect
4. #19128032C2semidirect
5. #19128031C2semidirect
6. #19128030C2semidirect
7. #19128029C2semidirect
8. #106×D83C2semidirect
9. #103×D83C2semidirect
11. #483×C83C2semidirect
30. #14581724C22semidirect
31. #14581723C22semidirect
32. #14581723C22semidirect
33. #14581717C22semidirect
34. #14581716C22semidirect
35. #14581715C22semidirect
40. #106×C83C22semidirect
41. #103×C83C22semidirect
141. D83#483direct (N × H)
143. C83C2×#483semidirect
151. #483D83direct (N × H)
178. #106C2×D83semidirect
182. #103C2×D83semidirect

Maximal subgroups

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

subgroupindex
C22⋊Q8×D832
C2×C4⋊C4×D832
#191280332
#191280322
#191280312
#191280302
#191280292
#106×D832
#103×D832
C22⋊C8×D832
#483×C832
C2×#48383