Finite Group Databasebeta

← #24918777

Subgroups — #24918777

Order 10624.

392 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 121 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#19128255C211yesmaximal, characteristic, complemented
2#19128249C211yesmaximal, characteristic
3#19127989C212yesmaximal, complemented
4#19127987C211yesmaximal, characteristic, complemented
5#19127986C211yesmaximal, characteristic, complemented
6#19127929C212yesmaximal, complemented
7#19127927C212yesmaximal, complemented
8#19127904C212yesmaximal, complemented
9#19127898C211yesmaximal, characteristic, complemented
10C4⋊C8×D83C211yesmaximal, characteristic, complemented
11#460×C83C211yesmaximal, characteristic, complemented
12C2×#13050589—21no—
13C2×#13050578C2212yes—
14C2×C8×D83C2212yes—
15#14581792C2211yescharacteristic
16#14581791C2211yescharacteristic
17#14581790C2211yescharacteristic
18#14581782—21no—
19#14581779—21no—
20#14581772C2211yescharacteristic
21#14581770—21no—
22#14581768C2211yescharacteristic
23#14581763—21no—
24#14581763C2212yes—
25C4×Dic166—21no—
26C4×Dic166C2211yescharacteristic
27D4×Dic83—21no—
28D4×Dic83C2212yes—
29#14581749—21no—
30C42×D83C2211yescharacteristic
31#14581724C2212yes—
32#14581723C2212yes—
33C4⋊(C83⋊C8)C2211yescharacteristic
34Dic83⋊C8C2212yes—
35#14581717C2212yes—
36#14581716C2212yes—
37#14581713C2212yes—
38C2×SD16×C83C2212yes—
39#127×C83C2211yescharacteristic
40C4×Q8×C83C2211yescharacteristic
41C4⋊C8×C83C2211yescharacteristic
42#102×C83C2212yes—
43#13050589—44no—
44C2×Dic166—21no—
45C2×Dic166—21no—
46C2×Dic166C2311yescharacteristic
47C2×C22⋊D83—22no—
48C22×Dic83—22no—
49C2×C4×D83D411yescharacteristic
50C2×C4×D83D411yescharacteristic
51C2×C4×D83C2311yescharacteristic
52#13050578—42no—
53C2×C83⋊C8C2312yes—
54C8×D83—42no—
55C4⋊Dic83—21no—
56C4⋊Dic83—21no—
57C4⋊Dic83C2312yes—
58C4⋊Dic83C2311yescharacteristic
59Dic83⋊C4—21no—
60Dic83⋊C4—21no—
61Dic83⋊C4—21no—
62Dic83⋊C4—21no—
63Dic83⋊C4—21no—
64C22⋊Dic83—22no—
65C22⋊Dic83—22no—
66(C2×D83)⋊C4—21no—
67(C2×D83)⋊C4—21no—
68C4×Dic83—21no—
69C4×Dic83D411yescharacteristic
70C4×Dic83D411yescharacteristic
71C4×Dic83C2311yescharacteristic
72C2×Q8×C83C2311yescharacteristic
73C2×D4×C83—21no—
74C2×D4×C83—21no—
75C2×D4×C83C2312yes—
76SD16×C83—42no—
77C2×C8×C83C2312yes—
78C4⋊C4×C83—21no—
79C4⋊C4×C83C2311yescharacteristic
80C42×C83—21no—
81C42×C83C2311yescharacteristic
82C23×C83—22no—
83C22×D83C2×D411yescharacteristic
84Q8×C83—21no—
85Q8×C83—21no—
86D4×C83—22no—
87D4×C83—22no—
88D4×C83—42no—
89D4×C83—42no—
90C2×C4×C83—21no—
91C2×C4×C83—21no—
92C2×C4×C83C2×D411yescharacteristic
93C2×C4×C83C2×D411yescharacteristic
94C2×C4×C83C2411yescharacteristic
95C22⋊D83—44no—
96C2×Dic83—21no—
97C2×Dic83—21no—
98C2×Dic83—22no—
99C2×Dic83—44no—
100C2×Dic83C2×D411yescharacteristic
101C2×Dic83C2×D411yescharacteristic
102C2×Dic83C2×D411yescharacteristic
103C4×D83—21no—
104C4×D83—21no—
105C4×D83—41no—
106C4×D83SD1614yescomplemented
107Dic166—21no—
108Dic166—21no—
109Dic166—42no—
110Dic166—42no—
111C8×C83—22no—
112C83⋊C8—22no—
113C22×C83—42no—
114C22×C83—42no—
115C22×C83—42no—
116C22×C83C22×D411yescharacteristic
117C2×D83—21no—
118C2×D83—21no—
119C2×D83C2×SD1612yes—
120C4×C83—21no—
121C4×C83—41no—
122C4×C83—41no—
123C4×C83C2×SD1612yes—
124C4×C83H(5,𝔽2)11yescharacteristic
125C4×C832−1+411yescharacteristic
126Dic83—21no—
127Dic83—21no—
128Dic83—41no—
129Dic83—42no—
130Dic83—41no—
131Dic83C2×SD1612yes—
132C2×C83—82no—
133C2×C83#53711yescharacteristic
134C2×C83C22×SD1611yescharacteristic
135C2×C83#57511yescharacteristic
136D83—22no—
137#3246—831nomaximal, Sylow
138C83#324611yescharacteristic, complemented, Sylow
139#533—831no—
140#532—831no—
141#476—832no—
142#475—831no—
143#475—831no—
144#465—832no—
145#465—832no—
146#461—832no—
147#460—831no—
148#460D8311yescharacteristic, complemented
149C2×C4⋊C8—831no—
150C2×#41—1661no—
151C2×SD16—832no—
152C2×SD16C2×D8312yes—
153C22×C8—832no—
154C4⋊Q8—831no—
155C4⋊Q8—831no—
156C4⋊Q8—831no—
157#127C2×D8311yescharacteristic
158#125—1661no—
159#124—1661no—
160C22⋊Q8—831no—
161C22⋊Q8—831no—
162C22⋊Q8—1661no—
163#121—832no—
164#121—1661no—
165C4×Q8—831no—
166C4×Q8—1661no—
167C4×Q8C2×D8311yescharacteristic
168C4×D4—832no—
169C4×D4—1661no—
170#117—1661no—
171C2×C42—831no—
172#106—832no—
173#106—832no—
174C4⋊C8—832no—
175C4⋊C8—831no—
176C4⋊C8C2×D8311yescharacteristic
177#103—832no—
178#103—832no—
179#102—832no—
180#102C2×D8312yes—
181#41—3324no—
182C2×Q8—831no—
183C2×Q8—1661no—
184C2×Q8—1661no—
185C2×Q8C22×D8311yescharacteristic
186C2×D4—21no—
187C2×D4—21no—
188C2×D4—1662no—
189C2×D4C22×D8312yes—
190C22×C4—831no—
191C22×C4—831no—
192C22×C4—831no—
193C22×C4—1662no—
194SD16—42no—
195SD16—3322no—
196C2×C8—3322no—
197C2×C8—832no—
198C2×C8C22×D8312yes—
199C4⋊C4—21no—
200C4⋊C4—831no—
201C4⋊C4—832no—
202C4⋊C4—1661no—
203C4⋊C4—1661no—
204C4⋊C4—1661no—
205C4⋊C4—1661no—
206C4⋊C4—1661no—
207C4⋊C4—1661no—
208C4⋊C4—1661no—
209C4⋊C4C22×D8311yescharacteristic
210C22⋊C4—1661no—
211C22⋊C4—1661no—
212C22⋊C4—1662no—
213C22⋊C4—1662no—
214C42—21no—
215C42—831no—
216C42—831no—
217C42—831no—
218C42—1661no—
219C42C22×D8311yescharacteristic
220C23—22no—
221C23—831no—
222Q8—21no—
223Q8—21no—
224Q8—3322no—
225Q8—3322no—
226Q8—1661no—
227Q8—1661no—
228D4—22no—
229D4—22no—
230D4—42no—
231D4—42no—
232D4—3324no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—3321no—
236C2×C4—3324no—
237C2×C4—831no—
238C2×C4—834no—
239C2×C4—831no—
240C2×C4—831no—
241C2×C4—1661no—
242C2×C4—1661no—
243C2×C4—1662no—
244C2×C4—1661no—
245C2×C4—1661no—
246C2×C4D4×D8311yescharacteristic
247C2×C4D4×D8311yescharacteristic
248C2×C4C23×D8311yescharacteristic
249C8—22no—
250C8—1662no—
251C22—42no—
252C22—42no—
253C22—42no—
254C22—832no—
255C22—1661no—
256C22—1661no—
257C22C2×D4×D8311yescharacteristic
258C4—21no—
259C4—41no—
260C4—41no—
261C4—3322no—
262C4—3321no—
263C4—3321no—
264C4—832no—
265C4—1661no—
266C4—1661no—
267C4SD16×D8312yes—
268C4#1458183711yescharacteristic
269C4#1458183911yescharacteristic
270C2—82no—
271C2—1662no—
272C2#1912828011yescharacteristic
273C2C2×SD16×D8311yescharacteristic
274C2#1912849711yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#460×C83C2
socleC22×C83C22×D4
prime radical#19128249C2
hypercenter#460D83
hypocenterC83#3246
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C42 ◁ #460

Complements

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

NH ≅ G/Ntype
1. #19128255C2semidirect
3. #19127989C2semidirect
4. #19127987C2semidirect
5. #19127986C2semidirect
6. #19127929C2semidirect
7. #19127927C2semidirect
8. #19127904C2semidirect
9. #19127898C2semidirect
10. C4⋊C8×D83C2semidirect
11. #460×C83C2semidirect
106. C4×D83SD16semidirect
138. C83#3246semidirect
148. #460D83semidirect

Maximal subgroups

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

subgroupindex
#191282552
#191282492
#191279892
#191279872
#191279862
#191279292
#191279272
#191279042
#191278982
C4⋊C8×D832
#460×C832
#324683