Finite Group Databasebeta

← #24918055

Subgroups — #24918055

Order 10624.

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

normal subgroups only

This group has 408 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#19128249C211yesmaximal, characteristic, complemented
2#19128230C211yesmaximal, characteristic, complemented
3#19128110C211yesmaximal, characteristic, complemented
4#19128065C211yesmaximal, characteristic, complemented
5#19128060C211yesmaximal, characteristic, complemented
6#19127951C212yesmaximal, complemented
7#19127944C212yesmaximal, complemented
8SD16×Dic83C212yesmaximal, complemented
9D8×Dic83C212yesmaximal, complemented
10#19127653C211yesmaximal, characteristic, complemented
11#496×C83C211yesmaximal, characteristic, complemented
12C2×#13050589—21no—
13C2×#13050573C2212yes—
14#14581792C2211yescharacteristic
15#14581785—21no—
16#14581784C2211yescharacteristic
17#14581779—21no—
18#14581774—21no—
19#14581768C2211yescharacteristic
20#14581763—21no—
21#14581763C2212yes—
22#14581763C2211yescharacteristic
23Q8×Dic83C2211yescharacteristic
24D4×Dic83—21no—
25D4×Dic83C2211yescharacteristic
26D4×Dic83C2212yes—
27(C22⋊D83)⋊C4—21no—
28#14581748—21no—
29C42×D83C2211yescharacteristic
30#14581726C2212yes—
31#14581724C2212yes—
32#14581716C2212yes—
33#14581713C2212yes—
34#14581713C2212yes—
35#14581713C2212yes—
36#14581703C2211yescharacteristic
37C8×Dic83C2212yes—
38C2×SD16×C83C2212yes—
39C2×D8×C83C2212yes—
40#127×C83C2211yescharacteristic
41#124×C83C2211yescharacteristic
42#97×C83C2211yescharacteristic
43#13050589—44no—
44C2×Dic166—21no—
45C2×C22⋊D83—22no—
46C2×C22⋊D83—21no—
47C22×Dic83—22no—
48C22×Dic83—21no—
49C2×C4×D83D411yescharacteristic
50C2×C4×D83D411yescharacteristic
51C2×C4×D83C2311yescharacteristic
52#13050573—24no—
53C2×C83⋊C8C2312yes—
54C4⋊Dic83—21no—
55C4⋊Dic83—21no—
56C4⋊Dic83C2311yescharacteristic
57C4⋊Dic83C2312yes—
58C4⋊Dic83C2311yescharacteristic
59Dic83⋊C4—21no—
60Dic83⋊C4—21no—
61Dic83⋊C4—21no—
62C22⋊Dic83—22no—
63C22⋊Dic83—22no—
64C22⋊Dic83—21no—
65C22⋊Dic83—21no—
66(C2×D83)⋊C4—21no—
67C4×Dic83—21no—
68C4×Dic83D411yescharacteristic
69C4×Dic83D411yescharacteristic
70C4×Dic83C2311yescharacteristic
71C2×Q8×C83C2311yescharacteristic
72C2×D4×C83—21no—
73C2×D4×C83—21no—
74C2×D4×C83C2311yescharacteristic
75C2×D4×C83C2312yes—
76SD16×C83—24no—
77D8×C83—24no—
78C2×C8×C83C2312yes—
79C22⋊C4×C83—21no—
80C22⋊C4×C83—21no—
81C42×C83C2311yescharacteristic
82C23×C83—21no—
83C23×C83—22no—
84C22×D83C2×D411yescharacteristic
85Q8×C83—22no—
86D4×C83—22no—
87D4×C83—22no—
88D4×C83—22no—
89D4×C83—42no—
90D4×C83—42no—
91C2×C4×C83—21no—
92C2×C4×C83C2×D411yescharacteristic
93C2×C4×C83C2×D411yescharacteristic
94C2×C4×C83C2411yescharacteristic
95C22⋊D83—42no—
96C22⋊D83—44no—
97C2×Dic83—21no—
98C2×Dic83—21no—
99C2×Dic83—22no—
100C2×Dic83—42no—
101C2×Dic83—44no—
102C2×Dic83C2×D411yescharacteristic
103C2×Dic83C2×D411yescharacteristic
104C2×Dic83C2×D411yescharacteristic
105C4×D83—22no—
106C4×D83—22no—
107C4×D83—41no—
108Dic166—42no—
109C8×C83#4114yescomplemented
110C83⋊C8—22no—
111C22×C83—41no—
112C22×C83—42no—
113C22×C83—42no—
114C22×C83—41no—
115C22×C83—41no—
116C22×C83—42no—
117C22×C83C22×D411yescharacteristic
118C2×D83—21no—
119C2×D83—21no—
120C2×D83—21no—
121C4×C83—21no—
122C4×C83—21no—
123C4×C83—41no—
124C4×C83C2×#4112yes—
125Dic83—21no—
126Dic83—21no—
127Dic83—42no—
128Dic83—41no—
129Dic83—41no—
130Dic83AGL(1,ℤ/8)12yes—
131C2×C83—82no—
132C2×C83—81no—
133C2×C83#53211yescharacteristic
134C2×C83C2×AGL(1,ℤ/8)11yescharacteristic
135C2×C83#57511yescharacteristic
136D83—41no—
137#3171—831nomaximal, Sylow
138C83#317111yescharacteristic, complemented, Sylow
139#532—831no—
140#529—831no—
141#501—831no—
142#496D8311yescharacteristic, complemented
143#490—831no—
144#489—831no—
145#469—832no—
146#468—832no—
147C4×SD16—832no—
148C4×D8—832no—
149C4×M16—831no—
150C2×#41—1661no—
151C2×SD16C2×D8312yes—
152C2×D8C2×D8312yes—
153C2×M16—832no—
154C4⋊Q8—831no—
155#127C2×D8311yescharacteristic
156#126—1661no—
157#125—831no—
158#124—1661no—
159#124C2×D8311yescharacteristic
160#123—1661no—
161C22⋊Q8—831no—
162#121—831no—
163#121—832no—
164#121—1661no—
165C4×Q8—831no—
166C4×D4—831no—
167C4×D4—832no—
168C4×D4—1661no—
169C4×D4—1661no—
170#117—1661no—
171C2×C42—831no—
172#107—832no—
173#106—832no—
174#103—832no—
175#102—832no—
176#102—832no—
177#102—832no—
178#97—831no—
179#97C2×D8311yescharacteristic
180C4×C8—832no—
181#41—3324no—
182C2×Q8—1661no—
183C2×Q8C22×D8311yescharacteristic
184C2×D4—21no—
185C2×D4—21no—
186C2×D4—1661no—
187C2×D4—1662no—
188C2×D4C22×D8311yescharacteristic
189C2×D4C22×D8312yes—
190C22×C4—831no—
191C22×C4—831no—
192C22×C4—831no—
193C22×C4—1661no—
194C22×C4—1662no—
195SD16—24no—
196D8—24no—
197M16—1664no—
198C2×C8—832no—
199C2×C8C22×D8312yes—
200C4⋊C4—831no—
201C4⋊C4—831no—
202C4⋊C4—832no—
203C4⋊C4—1661no—
204C4⋊C4—1661no—
205C4⋊C4—1661no—
206C4⋊C4—1661no—
207C4⋊C4—1661no—
208C22⋊C4—21no—
209C22⋊C4—21no—
210C22⋊C4—1661no—
211C22⋊C4—1661no—
212C22⋊C4—1662no—
213C22⋊C4—1661no—
214C22⋊C4—1662no—
215C42—831no—
216C42—831no—
217C42—831no—
218C42—1661no—
219C42C22×D8311yescharacteristic
220C23—21no—
221C23—22no—
222C23—831no—
223Q8—22no—
224Q8—3322no—
225D4—22no—
226D4—22no—
227D4—22no—
228D4—42no—
229D4—42no—
230D4—3324no—
231D4—3322no—
232C2×C4—21no—
233C2×C4—3321no—
234C2×C4—3324no—
235C2×C4—3322no—
236C2×C4—831no—
237C2×C4—831no—
238C2×C4—831no—
239C2×C4—1662no—
240C2×C4—1661no—
241C2×C4—1661no—
242C2×C4—1662no—
243C2×C4—1662no—
244C2×C4D4×D8311yescharacteristic
245C2×C4D4×D8311yescharacteristic
246C2×C4C23×D8311yescharacteristic
247C8—1662no—
248C8#1305058914yescomplemented
249C22—42no—
250C22—41no—
251C22—42no—
252C22—41no—
253C22—41no—
254C22—42no—
255C22—1661no—
256C22—1661no—
257C22—1661no—
258C22C2×D4×D8311yescharacteristic
259C4—21no—
260C4—21no—
261C4—41no—
262C4—3322no—
263C4—3321no—
264C4—3321no—
265C4—832no—
266C4—1661no—
267C4—1661no—
268C4C2×#1305058912yes—
269C2—82no—
270C2—81no—
271C2—3321no—
272C2#1912824911yescharacteristic
273C2AGL(1,ℤ/8)×D8311yescharacteristic
274C2#1912849711yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#496×C83C2
socleC22×C83C22×D4
hypercenter#496D83
hypocenterC83#3171
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C42 ◁ #496

Complements

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

NH ≅ G/Ntype
1. #19128249C2semidirect
2. #19128230C2semidirect
3. #19128110C2semidirect
4. #19128065C2semidirect
5. #19128060C2semidirect
6. #19127951C2semidirect
7. #19127944C2semidirect
8. SD16×Dic83C2semidirect
9. D8×Dic83C2semidirect
10. #19127653C2semidirect
11. #496×C83C2semidirect
109. C8×C83#41semidirect
138. C83#3171semidirect
142. #496D83semidirect
248. C8#13050589semidirect

Maximal subgroups

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

subgroupindex
#191282492
#191282302
#191281102
#191280652
#191280602
#191279512
#191279442
SD16×Dic832
D8×Dic832
#191276532
#496×C832
#317183