Finite Group Databasebeta

← #24918323

Subgroups — #24918323

Order 10624.

442 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 442 conjugacy classes of subgroups — too many to draw. Only its 109 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#19128267C212yesmaximal, complemented
2#19128028C212yesmaximal, complemented
3#19128023C212yesmaximal, complemented
4#19127952C212yesmaximal, complemented
5#19127937C211yesmaximal, characteristic, complemented
6#19127836C212yesmaximal, complemented
7#19127833C212yesmaximal, complemented
8#19127667C211yesmaximal, characteristic, complemented
9#466×C83C211yesmaximal, characteristic, complemented
10C2×#13050590C2211yescharacteristic
11C2×D4×D83C2212yescomplemented
12C2×D664C2211yescharacteristic
13C2×#13050575C2212yes—
14C2×#13050574C2211yescharacteristic, complemented
15#14581788—22no—
16#14581778—22no—
17#14581767C2212yes—
18#14581766C2212yescomplemented
19#14581763C2212yes—
20#14581762—22no—
21#14581760—22no—
22#14581750C2212yescomplemented
23#14581726C2211yescharacteristic, complemented
24#14581723C2212yescomplemented
25#14581714C2212yescomplemented
26#14581713C2212yescomplemented
27#14581712C2212yescomplemented
28(C2×D83)⋊C8C2212yescomplemented
29(C2×D83)⋊C8C2212yescomplemented
30C2×D8×C83C2211yescharacteristic
31C22×C8×C83C2211yescharacteristic, complemented
32#121×C83C2212yescomplemented
33#107×C83C2211yescharacteristic, complemented
34#102×C83C2212yescomplemented
35C23×D83—22no—
36#13050590—41no—
37#13050590—41no—
38C2×Dic166D411yescharacteristic
39C2×D332—22no—
40C2×D332D411yescharacteristic
41C2×D332C2312yes—
42C2×C22⋊D83—22no—
43C2×C22⋊D83—22no—
44C2×C22⋊D83—22no—
45C2×C22⋊D83D412yes—
46D4×D83—42no—
47D4×D83—42no—
48C2×C4×D83C2312yes—
49D664—41no—
50#13050575—42no—
51#13050574—41no—
52C2×C83⋊C8C2312yes—
53C4⋊Dic83C2312yes—
54C22⋊Dic83—22no—
55(C2×D83)⋊C4—22no—
56(C2×D83)⋊C4—22no—
57(C2×D83)⋊C4—22no—
58C4×Dic83—22no—
59C2×D4×C83—22no—
60C2×D4×C83C2312yes—
61C22×C4×C83C2311yescharacteristic
62D8×C83—41no—
63C2×C8×C83—41no—
64C2×C8×C83C2311yescharacteristic
65C2×C8×C83C2311yescharacteristic
66C4⋊C4×C83C2312yes—
67C22⋊C4×C83—22no—
68C23×C83—22no—
69C23×C83C2×D411yescharacteristic
70C22×D83—22no—
71C22×D83—44no—
72C22×D83—82no—
73C22×D83—82no—
74C22×D83C2×D412yes—
75D4×C83—22no—
76D4×C83—22no—
77D4×C83—82no—
78C2×C4×C83—21no—
79C2×C4×C83—21no—
80C2×C4×C83—22no—
81C2×C4×C83C2×D411yescharacteristic
82C2×C4×C83C2411yescharacteristic
83C22⋊D83—42no—
84C22⋊D83—44no—
85C22⋊D83—82no—
86C22⋊D83—82no—
87C2×Dic83—22no—
88C2×Dic83C2×D412yes—
89D332—22no—
90D332—22no—
91D332—41no—
92D332—82no—
93C4×D83—22no—
94C4×D83—22no—
95Dic166—41no—
96C8×C83—21no—
97C8×C83—21no—
98C83⋊C8—22no—
99C22×C83—21no—
100C22×C83—21no—
101C22×C83—21no—
102C22×C83—42no—
103C22×C83—42no—
104C22×C83—42no—
105C22×C83C22×D411yescharacteristic
106C2×D83—22no—
107C2×D83—22no—
108C2×D83—22no—
109C2×D83—42no—
110C2×D83—42no—
111C2×D83—42no—
112C2×D83—88no—
113C4×C83—21no—
114C4×C83—42no—
115C4×C83H(5,𝔽2)11yescharacteristic
116C4×C83H(5,𝔽2)11yescharacteristic
117Dic83—22no—
118Dic83—42no—
119C2×C83—41no—
120C2×C83—82no—
121C2×C83#53411yescharacteristic
122C2×C83#57611yescharacteristic
123C2×C83#57611yescharacteristic
124D83—42no—
125D83—82no—
126#3199—831nomaximal, Sylow
127C83#319911yescharacteristic, complemented, Sylow
128#535—832no—
129#482—832no—
130#482—832no—
131#469—832no—
132#466—831no—
133#466D8311yescharacteristic, complemented
134#447—832no—
135#447—832no—
136#408—831no—
137C2×#41—831no—
138C22×D4—832no—
139C2×D8—832no—
140C2×D8—831no—
141C2×D8C2×D8311yescharacteristic
142C2×M16—831no—
143C22×C8C2×D8311yescharacteristic, complemented
144#127—1662no—
145#124—1662no—
146#121—832no—
147#121—832no—
148#121—832no—
149#121C2×D8312yescomplemented
150#120—1662no—
151#120—1662no—
152#117—832no—
153#107—831no—
154#107C2×D8311yescharacteristic, complemented
155#106—832no—
156#102—832no—
157#102—832no—
158#102—832no—
159#102C2×D8312yescomplemented
160C22⋊C8—832no—
161C22⋊C8—832no—
162C24—1662no—
163#41—3321no—
164#41—3321no—
165C2×Q8—831no—
166C2×D4—22no—
167C2×D4—3322no—
168C2×D4—3322no—
169C2×D4—832no—
170C2×D4—831no—
171C2×D4—832no—
172C2×D4—1662no—
173C2×D4—1662no—
174C2×D4—1662no—
175C2×D4—1662no—
176C2×D4C22×D8312yes—
177C22×C4—832no—
178C22×C4C22×D8311yescharacteristic
179D8—41no—
180D8—3321no—
181D8—3322no—
182M16—3321no—
183C2×C8—41no—
184C2×C8—832no—
185C2×C8C22×D8311yescharacteristic
186C2×C8C22×D8311yescharacteristic
187C4⋊C4—832no—
188C4⋊C4C22×D8312yes—
189C22⋊C4—22no—
190C22⋊C4—1662no—
191C22⋊C4—1662no—
192C22⋊C4—1662no—
193C22⋊C4—1662no—
194C42—1662no—
195C23—22no—
196C23—3324no—
197C23—832no—
198C23—6642no—
199C23—6642no—
200C23—1662no—
201C23D4×D8311yescharacteristic
202Q8—3321no—
203D4—22no—
204D4—22no—
205D4—82no—
206D4—3321no—
207D4—3322no—
208D4—3324no—
209D4—6642no—
210D4—6642no—
211D4—6642no—
212D4—1662no—
213D4—1662no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—22no—
217C2×C4—832no—
218C2×C4—1662no—
219C2×C4—1662no—
220C2×C4—1662no—
221C2×C4D4×D8311yescharacteristic
222C2×C4C23×D8311yescharacteristic
223C8—21no—
224C8—21no—
225C8—1662no—
226C22—21no—
227C22—21no—
228C22—21no—
229C22—42no—
230C22—42no—
231C22—42no—
232C22—3322no—
233C22—3322no—
234C22—3322no—
235C22—6648no—
236C22—1662no—
237C22—1662no—
238C22—1662no—
239C22C2×D4×D8311yescharacteristic
240C4—21no—
241C4—42no—
242C4—3322no—
243C4—1662no—
244C4#1458183711yescharacteristic
245C4#1458183811yescharacteristic
246C2—41no—
247C2—82no—
248C2—3322no—
249C2—6642no—
250C2#1912825911yescharacteristic
251C2#1912850211yescharacteristic
252C2#1912850311yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#466×C83C2
socleC22×C83C22×D4
hypercenter#466D83
hypocenterC83#3199
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ #466

Complements

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

NH ≅ G/Ntype
1. #19128267C2semidirect
2. #19128028C2semidirect
3. #19128023C2semidirect
4. #19127952C2semidirect
5. #19127937C2semidirect
6. #19127836C2semidirect
7. #19127833C2semidirect
8. #19127667C2semidirect
9. #466×C83C2semidirect
11. C2×D4×D83C22semidirect
14. C2×#13050574C22semidirect
18. #14581766C22semidirect
22. #14581750C22semidirect
23. #14581726C22semidirect
24. #14581723C22semidirect
25. #14581714C22semidirect
26. #14581713C22semidirect
27. #14581712C22semidirect
28. (C2×D83)⋊C8C22semidirect
29. (C2×D83)⋊C8C22semidirect
31. C22×C8×C83C22semidirect
32. #121×C83C22semidirect
33. #107×C83C22semidirect
34. #102×C83C22semidirect
127. C83#3199semidirect
133. #466D83semidirect
143. C22×C8C2×D83semidirect
149. #121C2×D83semidirect
154. #107C2×D83semidirect
159. #102C2×D83semidirect

Maximal subgroups

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

subgroupindex
#191282672
#191280282
#191280232
#191279522
#191279372
#191278362
#191278332
#191276672
#466×C832
#319983