Finite Group Databasebeta

← #24918791

Subgroups — #24918791

Order 10624.

398 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 115 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#19128252C211yesmaximal, characteristic, complemented
2#19128233C211yesmaximal, characteristic, complemented
3#19128000C212yesmaximal, complemented
4#19127993C211yesmaximal, characteristic, complemented
5#19127936C212yesmaximal, complemented
6#19127931C212yesmaximal, complemented
7#19127915C212yesmaximal, complemented
8#19127902C211yesmaximal, characteristic, complemented
9#19127896C211yesmaximal, characteristic, complemented
10C4⋊C8×D83C211yesmaximal, characteristic, complemented
11#476×C83C211yesmaximal, characteristic, complemented
12C2×#13050591—21no—
13C2×#13050579C2212yes—
14C2×#13050575C2212yes—
15C2×C8×D83C2212yes—
16#14581789C2211yescharacteristic
17#14581785—21no—
18#14581783C2211yescharacteristic
19#14581781—21no—
20#14581779C2211yescharacteristic
21#14581774—21no—
22#14581772C2211yescharacteristic
23#14581767—21no—
24#14581767C2212yes—
25#14581763C2211yescharacteristic
26C4×Dic166C2211yescharacteristic
27#14581754—21no—
28#14581754C2212yes—
29C4×C22⋊D83—21no—
30#14581751—21no—
31C42×D83C2211yescharacteristic
32#14581725C2212yes—
33C4⋊(C83⋊C8)C2211yescharacteristic
34Dic83⋊C8C2212yes—
35#14581715C2212yes—
36#14581714C2212yes—
37#14581712C2212yes—
38C4⋊Q8×C83C2211yescharacteristic
39C4×D4×C83C2211yescharacteristic
40#106×C83C2212yes—
41C4⋊C8×C83C2211yescharacteristic
42#102×C83C2212yes—
43#13050591—44no—
44C2×Dic166C2311yescharacteristic
45C2×D332—22no—
46C2×D332—21no—
47C2×D332—21no—
48C2×D332C2312yes—
49C2×C22⋊D83—21no—
50C2×C4×D83—22no—
51C2×C4×D83D411yescharacteristic
52C2×C4×D83D411yescharacteristic
53C2×C4×D83C2311yescharacteristic
54#13050579—42no—
55#13050575—42no—
56C2×C83⋊C8C2312yes—
57C8×D83—42no—
58C4⋊Dic83—21no—
59C4⋊Dic83—21no—
60C4⋊Dic83C2311yescharacteristic
61Dic83⋊C4—21no—
62Dic83⋊C4—21no—
63C22⋊Dic83—21no—
64C22⋊Dic83—21no—
65C22⋊Dic83—21no—
66(C2×D83)⋊C4—21no—
67(C2×D83)⋊C4—22no—
68(C2×D83)⋊C4—22no—
69C4×Dic83—21no—
70C4×Dic83D411yescharacteristic
71C4×Dic83D411yescharacteristic
72C4×Dic83C2311yescharacteristic
73C2×Q8×C83—21no—
74C2×D4×C83C2311yescharacteristic
75C22×C4×C83—21no—
76C2×C8×C83C2312yes—
77C4⋊C4×C83—21no—
78C4⋊C4×C83C2312yes—
79C4⋊C4×C83C2311yescharacteristic
80C22⋊C4×C83—21no—
81C42×C83C2311yescharacteristic
82C23×C83—21no—
83C22×D83—22no—
84C22×D83C2×D411yescharacteristic
85Q8×C83—42no—
86D4×C83—21no—
87D4×C83—21no—
88C2×C4×C83—21no—
89C2×C4×C83—22no—
90C2×C4×C83—42no—
91C2×C4×C83C2×D411yescharacteristic
92C2×C4×C83C2×D411yescharacteristic
93C2×C4×C83C2411yescharacteristic
94C22⋊D83—42no—
95C2×Dic83—21no—
96C2×Dic83—21no—
97C2×Dic83C2×D411yescharacteristic
98C2×Dic83C2×D411yescharacteristic
99C2×Dic83C2×D411yescharacteristic
100D332—22no—
101D332—22no—
102D332—42no—
103D332—44no—
104D332—42no—
105C4×D83—22no—
106C4×D83—22no—
107C4×D83—21no—
108C4×D83—21no—
109C4×D83—44no—
110Dic166—21no—
111Dic166—21no—
112C8×C83—22no—
113C83⋊C8—22no—
114C22×C83—41no—
115C22×C83—41no—
116C22×C83—41no—
117C22×C83C22×D411yescharacteristic
118C2×D83—21no—
119C2×D83—21no—
120C2×D83—42no—
121C2×D83—42no—
122C2×D83—42no—
123C2×D83AGL(1,ℤ/8)12yes—
124C4×C83—21no—
125C4×C83—41no—
126C4×C83—42no—
127C4×C83#13512yes—
128C4×C83H(5,𝔽2)11yescharacteristic
129C4×C832−1+411yescharacteristic
130Dic83—21no—
131Dic83—21no—
132Dic83—21no—
133Dic83—41no—
134Dic83—41no—
135C2×C83—81no—
136C2×C83#53711yescharacteristic
137C2×C83C2×#13511yescharacteristic
138C2×C83C2×AGL(1,ℤ/8)11yescharacteristic
139D83—22no—
140D83—82no—
141#3247—831nomaximal, Sylow
142C83#324711yescharacteristic, complemented, Sylow
143#532—831no—
144#529—831no—
145#478—832no—
146#477—831no—
147#476D8311yescharacteristic, complemented
148#466—832no—
149#465—832no—
150#463—832no—
151#460—831no—
152#459—831no—
153C2×C4⋊C8—831no—
154C2×#41—1661no—
155C2×SD16—832no—
156C2×D8—832no—
157C22×C8—832no—
158C4⋊Q8C2×D8311yescharacteristic
159#127—831no—
160#126—1661no—
161#125—831no—
162#124—831no—
163#124—1661no—
164#123—1661no—
165C22⋊Q8—831no—
166#121—831no—
167#121—832no—
168#121—1661no—
169C4×Q8—831no—
170C4×D4—832no—
171C4×D4—1661no—
172C4×D4—1661no—
173C4×D4C2×D8311yescharacteristic
174#117—1661no—
175C2×C42—831no—
176#107—832no—
177#106C2×D8312yes—
178C4⋊C8—832no—
179C4⋊C8—831no—
180C4⋊C8C2×D8311yescharacteristic
181#103—832no—
182#102—832no—
183#102—832no—
184#102C2×D8312yes—
185#41—3324no—
186C2×Q8—21no—
187C2×Q8—831no—
188C2×D4—832no—
189C2×D4—1662no—
190C2×D4—1661no—
191C2×D4—1661no—
192C2×D4—1661no—
193C2×D4C22×D8311yescharacteristic
194C22×C4—21no—
195C22×C4—831no—
196C22×C4—831no—
197C22×C4—831no—
198C22×C4—1662no—
199SD16—3322no—
200D8—3322no—
201C2×C8—3322no—
202C2×C8—832no—
203C2×C8C22×D8312yes—
204C4⋊C4—21no—
205C4⋊C4—831no—
206C4⋊C4—1661no—
207C4⋊C4—1661no—
208C4⋊C4—1661no—
209C4⋊C4—1661no—
210C4⋊C4C22×D8311yescharacteristic
211C4⋊C4C22×D8312yes—
212C22⋊C4—21no—
213C22⋊C4—1662no—
214C22⋊C4—1662no—
215C22⋊C4—1661no—
216C22⋊C4—1661no—
217C22⋊C4—1661no—
218C22⋊C4—1661no—
219C42—831no—
220C42—831no—
221C42—831no—
222C42—1661no—
223C42C22×D8311yescharacteristic
224C23—21no—
225C23—831no—
226C23—1662no—
227Q8—42no—
228Q8—1661no—
229Q8—1661no—
230D4—21no—
231D4—21no—
232D4—3322no—
233D4—3324no—
234D4—3322no—
235D4—3322no—
236D4—1662no—
237D4—1662no—
238C2×C4—21no—
239C2×C4—22no—
240C2×C4—42no—
241C2×C4—3324no—
242C2×C4—831no—
243C2×C4—831no—
244C2×C4—831no—
245C2×C4—1662no—
246C2×C4—1661no—
247C2×C4—1661no—
248C2×C4—1662no—
249C2×C4—1661no—
250C2×C4—1661no—
251C2×C4D4×D8311yescharacteristic
252C2×C4D4×D8311yescharacteristic
253C2×C4C23×D8311yescharacteristic
254C8—22no—
255C8—1662no—
256C22—41no—
257C22—41no—
258C22—41no—
259C22—3322no—
260C22—3322no—
261C22—3322no—
262C22—832no—
263C22—1661no—
264C22—1661no—
265C22C2×D4×D8311yescharacteristic
266C4—21no—
267C4—41no—
268C4—42no—
269C4—3321no—
270C4—3321no—
271C4—1661no—
272C4—1661no—
273C4—1661no—
274C4#1458181312yes—
275C4#1458183811yescharacteristic
276C4#1458184011yescharacteristic
277C2—81no—
278C2—6642no—
279C2—1662no—
280C2#1912828111yescharacteristic
281C2C2×#1458181311yescharacteristic
282C2AGL(1,ℤ/8)×D8311yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#476×C83C2
socleC22×C83C22×D4
hypercenter#476D83
hypocenterC83#3247
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C42 ◁ #476

Complements

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

NH ≅ G/Ntype
1. #19128252C2semidirect
2. #19128233C2semidirect
3. #19128000C2semidirect
4. #19127993C2semidirect
5. #19127936C2semidirect
6. #19127931C2semidirect
7. #19127915C2semidirect
8. #19127902C2semidirect
9. #19127896C2semidirect
10. C4⋊C8×D83C2semidirect
11. #476×C83C2semidirect
142. C83#3247semidirect
147. #476D83semidirect

Maximal subgroups

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

subgroupindex
#191282522
#191282332
#191280002
#191279932
#191279362
#191279312
#191279152
#191279022
#191278962
C4⋊C8×D832
#476×C832
#324783