Finite Group Databasebeta

← #24918781

Subgroups — #24918781

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#19128256C211yesmaximal, characteristic, complemented
2#19128248C211yesmaximal, characteristic
3#19127988C211yesmaximal, characteristic, complemented
4#19127985C212yesmaximal, complemented
5#19127931C212yesmaximal, complemented
6#19127929C212yesmaximal, complemented
7#19127907C211yesmaximal, characteristic, complemented
8#19127906C211yesmaximal, characteristic, complemented
9#19127900C212yesmaximal, complemented
10C4⋊C8×D83C211yesmaximal, characteristic, complemented
11#476×C83C211yesmaximal, characteristic, complemented
12C2×#13050590—21no—
13C2×#13050579C2212yes—
14C2×#13050578C2212yes—
15C2×C8×D83C2212yes—
16#14581792C2211yescharacteristic
17#14581791C2211yescharacteristic
18#14581788C2211yescharacteristic
19#14581783—21no—
20#14581778—21no—
21#14581772C2212yes—
22#14581768—21no—
23#14581767C2211yescharacteristic
24#14581765—21no—
25#14581763C2211yescharacteristic
26Q8×Dic83—21no—
27#14581757C2212yes—
28C4×D332C2211yescharacteristic
29C4×C22⋊D83—21no—
30#14581750—21no—
31C42×D83C2211yescharacteristic
32#14581723C2212yes—
33C4⋊(C83⋊C8)C2211yescharacteristic
34Dic83⋊C8C2212yes—
35#14581717C2212yes—
36#14581715C2212yes—
37#14581712C2212yes—
38C4⋊Q8×C83C2211yescharacteristic
39C4×D4×C83C2211yescharacteristic
40#106×C83C2212yes—
41C4⋊C8×C83C2211yescharacteristic
42#102×C83C2212yes—
43#13050590—44no—
44C2×Dic166—21no—
45C2×Dic166C2312yes—
46C2×D332—21no—
47C2×D332C2311yescharacteristic
48C2×C22⋊D83—21no—
49C2×C22⋊D83—21no—
50C2×C22⋊D83—21no—
51C2×C4×D83—21no—
52C2×C4×D83D411yescharacteristic
53C2×C4×D83D411yescharacteristic
54C2×C4×D83C2311yescharacteristic
55#13050579—42no—
56#13050578—42no—
57C2×C83⋊C8C2312yes—
58C8×D83—42no—
59C4⋊Dic83—21no—
60C4⋊Dic83—21no—
61C4⋊Dic83—22no—
62C4⋊Dic83C2311yescharacteristic
63Dic83⋊C4—21no—
64Dic83⋊C4—22no—
65C22⋊Dic83—21no—
66(C2×D83)⋊C4—22no—
67(C2×D83)⋊C4—21no—
68(C2×D83)⋊C4—21no—
69C4×Dic83—22no—
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×C83C2311yescharacteristic
79C4⋊C4×C83C2312yes—
80C22⋊C4×C83—21no—
81C42×C83C2311yescharacteristic
82C23×C83—21no—
83C22×D83—21no—
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—
95C22⋊D83—42no—
96C22⋊D83—42no—
97C2×Dic83—22no—
98C2×Dic83—21no—
99C2×Dic83C2×D411yescharacteristic
100C2×Dic83C2×D411yescharacteristic
101C2×Dic83C2×D411yescharacteristic
102D332—21no—
103D332—21no—
104D332—42no—
105C4×D83—21no—
106C4×D83—21no—
107C4×D83—41no—
108C4×D83—42no—
109C4×D83SD1614yescomplemented
110Dic166—22no—
111Dic166—22no—
112Dic166—42no—
113C8×C83—22no—
114C83⋊C8—22no—
115C22×C83—41no—
116C22×C83—41no—
117C22×C83—41no—
118C22×C83C22×D411yescharacteristic
119C2×D83—21no—
120C2×D83—21no—
121C2×D83—41no—
122C2×D83—41no—
123C2×D83—41no—
124C2×D83C2×SD1612yes—
125C4×C83—21no—
126C4×C83—41no—
127C4×C83—42no—
128C4×C83C2×SD1612yes—
129C4×C83H(5,𝔽2)11yescharacteristic
130C4×C832−1+411yescharacteristic
131Dic83—21no—
132Dic83—21no—
133Dic83—42no—
134Dic83—41no—
135Dic83C2×SD1612yes—
136C2×C83—81no—
137C2×C83#53711yescharacteristic
138C2×C83C22×SD1611yescharacteristic
139C2×C83#57511yescharacteristic
140D83—22no—
141D83—81no—
142#3246—831nomaximal, Sylow
143C83#324611yescharacteristic, complemented, Sylow
144#533—831no—
145#532—831no—
146#476—831no—
147#476D8311yescharacteristic, complemented
148#475—832no—
149#465—832no—
150#465—832no—
151#461—831no—
152#461—831no—
153#460—832no—
154C2×C4⋊C8—831no—
155C2×#41—1661no—
156C2×SD16—832no—
157C2×SD16—832no—
158C22×C8—832no—
159C4⋊Q8—831no—
160C4⋊Q8—831no—
161C4⋊Q8C2×D8311yescharacteristic
162#127—831no—
163#125—1661no—
164#124—1661no—
165C22⋊Q8—832no—
166C22⋊Q8—1661no—
167#121—831no—
168#121—831no—
169#121—1661no—
170C4×Q8—832no—
171C4×Q8—1661no—
172C4×D4—831no—
173C4×D4—1661no—
174C4×D4C2×D8311yescharacteristic
175#117—1661no—
176C2×C42—831no—
177#106—832no—
178#106C2×D8312yes—
179C4⋊C8—832no—
180C4⋊C8—831no—
181C4⋊C8C2×D8311yescharacteristic
182#103—832no—
183#103—832no—
184#102—832no—
185#102C2×D8312yes—
186#41—3324no—
187C2×Q8—21no—
188C2×Q8—832no—
189C2×Q8—1661no—
190C2×D4—831no—
191C2×D4—1661no—
192C2×D4—1661no—
193C2×D4—1661no—
194C2×D4—1661no—
195C2×D4C22×D8311yescharacteristic
196C22×C4—21no—
197C22×C4—831no—
198C22×C4—831no—
199C22×C4—831no—
200C22×C4—1661no—
201SD16—3322no—
202SD16—3322no—
203C2×C8—3322no—
204C2×C8—832no—
205C2×C8C22×D8312yes—
206C4⋊C4—21no—
207C4⋊C4—831no—
208C4⋊C4—1661no—
209C4⋊C4—1662no—
210C4⋊C4—1662no—
211C4⋊C4—1661no—
212C4⋊C4—1661no—
213C4⋊C4C22×D8312yes—
214C4⋊C4C22×D8311yescharacteristic
215C22⋊C4—21no—
216C22⋊C4—1661no—
217C22⋊C4—1662no—
218C22⋊C4—1661no—
219C22⋊C4—1661no—
220C42—831no—
221C42—831no—
222C42—831no—
223C42—1662no—
224C42C22×D8311yescharacteristic
225C23—21no—
226C23—831no—
227C23—1661no—
228Q8—42no—
229Q8—3322no—
230Q8—1662no—
231Q8—1662no—
232D4—21no—
233D4—21no—
234D4—3322no—
235D4—3322no—
236D4—3322no—
237D4—3322no—
238D4—1661no—
239D4—1661no—
240C2×C4—22no—
241C2×C4—21no—
242C2×C4—42no—
243C2×C4—3321no—
244C2×C4—3322no—
245C2×C4—831no—
246C2×C4—834no—
247C2×C4—831no—
248C2×C4—831no—
249C2×C4—1662no—
250C2×C4—1661no—
251C2×C4—1661no—
252C2×C4—1661no—
253C2×C4D4×D8311yescharacteristic
254C2×C4D4×D8311yescharacteristic
255C2×C4C23×D8311yescharacteristic
256C8—22no—
257C8—1662no—
258C22—41no—
259C22—41no—
260C22—41no—
261C22—3321no—
262C22—3321no—
263C22—3321no—
264C22—832no—
265C22—1661no—
266C22—1661no—
267C22C2×D4×D8311yescharacteristic
268C4—21no—
269C4—41no—
270C4—42no—
271C4—3321no—
272C4—3322no—
273C4—832no—
274C4—1661no—
275C4—1661no—
276C4SD16×D8312yes—
277C4#1458183811yescharacteristic
278C4#1458184011yescharacteristic
279C2—81no—
280C2—6641no—
281C2—1662no—
282C2#1912828111yescharacteristic
283C2C2×SD16×D8311yescharacteristic
284C2#1912849711yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#476×C83C2
socleC22×C83C22×D4
prime radical#19128248C2
hypercenter#476D83
hypocenterC83#3246
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. #19128256C2semidirect
3. #19127988C2semidirect
4. #19127985C2semidirect
5. #19127931C2semidirect
6. #19127929C2semidirect
7. #19127907C2semidirect
8. #19127906C2semidirect
9. #19127900C2semidirect
10. C4⋊C8×D83C2semidirect
11. #476×C83C2semidirect
109. C4×D83SD16semidirect
143. C83#3246semidirect
147. #476D83semidirect

Maximal subgroups

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

subgroupindex
#191282562
#191282482
#191279882
#191279852
#191279312
#191279292
#191279072
#191279062
#191279002
C4⋊C8×D832
#476×C832
#324683