Finite Group Databasebeta

← #24918382

Subgroups — #24918382

Order 10624.

532 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 532 conjugacy classes of subgroups — too many to draw. Only its 125 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#127×D83C211yesmaximal, characteristic, complemented
3#19128078C211yesmaximal, characteristic, complemented
4#19128077C211yesmaximal, characteristic, complemented
5#19128049C211yesmaximal, characteristic, complemented
6#19127895C214yesmaximal, complemented
7#19127836C214yesmaximal, complemented
8#19127752C211yesmaximal, characteristic, complemented
9#486×C83C211yesmaximal, characteristic, complemented
10C2×#13050591—21no—
11C2×D4×D83—21no—
12C2×D4×D83—21no—
13C2×D4×D83C2212yes—
14C2×D664C2214yescomplemented
15C2×#13050575C2214yescomplemented
16#14581789C2211yescharacteristic
17#14581789C2211yescharacteristic, complemented
18#14581788—21no—
19#14581788—21no—
20#14581788C2212yes—
21#14581781—21no—
22#14581767—21no—
23#14581767C2212yes—
24#14581754—21no—
25#14581754C2212yescomplemented
26C42×D83C2211yescharacteristic
27Dic83⋊C8C2212yescomplemented
28#14581712C2214yescomplemented
29(C2×D83)⋊C8C2212yes—
30C4×C83⋊C8C2211yescharacteristic, complemented
31C4⋊Q8×C83C2211yescharacteristic, complemented
32#127×C83C2211yescharacteristic
33#102×C83C2214yescomplemented
34C4×C8×C83C2211yescharacteristic, complemented
35C23×D83—22no—
36#13050591—44no—
37C2×D332—22no—
38C2×D332—21no—
39C2×D332—21no—
40C2×D332—21no—
41C2×D332—21no—
42C2×D332C2312yes—
43C2×D332C2312yes—
44C2×C22⋊D83—22no—
45C2×C22⋊D83—22no—
46C2×C22⋊D83—22no—
47D4×D83—44no—
48D4×D83—44no—
49D4×D83—44no—
50C2×C4×D83—22no—
51C2×C4×D83D411yescharacteristic
52C2×C4×D83D411yescharacteristic
53C2×C4×D83C2311yescharacteristic
54D664—44no—
55#13050575—44no—
56C2×C83⋊C8C2312yes—
57(C2×D83)⋊C4—22no—
58(C2×D83)⋊C4—22no—
59C4×Dic83D411yescharacteristic
60C4×Dic83D411yescharacteristic
61C4×Dic83C2311yescharacteristic
62C2×Q8×C83—21no—
63C2×D4×C83—21no—
64C2×D4×C83—21no—
65C2×D4×C83C2312yes—
66C2×C8×C83C2312yes—
67C4⋊C4×C83—21no—
68C4⋊C4×C83C2312yes—
69C42×C83C2311yescharacteristic
70C23×C83—22no—
71C22×D83—22no—
72C22×D83—22no—
73C22×D83—44no—
74C22×D83—44no—
75C22×D83—44no—
76C22×D83C2×D411yescharacteristic
77Q8×C83—42no—
78D4×C83—24no—
79D4×C83—42no—
80D4×C83—42no—
81C2×C4×C83—22no—
82C2×C4×C83C2×D411yescharacteristic
83C2×C4×C83C2×D411yescharacteristic
84C2×C4×C83C2411yescharacteristic
85C22⋊D83—44no—
86C22⋊D83—44no—
87C22⋊D83—44no—
88C2×Dic83C2×D411yescharacteristic
89C2×Dic83C2×D411yescharacteristic
90C2×Dic83C2×D411yescharacteristic
91D332—24no—
92D332—24no—
93D332—44no—
94D332—42no—
95D332—42no—
96D332—42no—
97D332—42no—
98C4×D83—22no—
99C4×D83—22no—
100C4×D83—44no—
101C4×D83D814yescomplemented
102C8×C83—22no—
103C83⋊C8—22no—
104C22×C83—42no—
105C22×C83—42no—
106C22×C83—42no—
107C22×C83C22×D411yescharacteristic
108C2×D83—21no—
109C2×D83—21no—
110C2×D83—42no—
111C2×D83—42no—
112C2×D83—42no—
113C2×D83—42no—
114C2×D83—42no—
115C2×D83—42no—
116C2×D83—88no—
117C2×D83C2×D812yes—
118C4×C83—42no—
119C4×C83C2×D812yes—
120C4×C83AGL(1,ℤ/8)12yes—
121C4×C83H(5,𝔽2)12yes—
122Dic83—21no—
123Dic83—21no—
124Dic83C2×D812yes—
125C2×C83—82no—
126C2×C83#53511yescharacteristic
127C2×C83C22×D811yescharacteristic
128C2×C83C2×AGL(1,ℤ/8)11yescharacteristic
129D83—22no—
130D83—82no—
131D83—82no—
132#3205—831nomaximal, Sylow
133C83#320511yescharacteristic, complemented, Sylow
134#532—831no—
135C2×#127—831no—
136#493—831no—
137#493—831no—
138#486—831no—
139#486D8311yescharacteristic, complemented
140#459—834no—
141#447—834no—
142#431—831no—
143C2×#41—1661no—
144C22×D4—832no—
145C22×D4—1661no—
146C22×D4—1661no—
147C2×D8—834no—
148C2×D8—834no—
149C4⋊Q8C2×D8311yescharacteristic, complemented
150#127—832no—
151#127—831no—
152#127—831no—
153#127—1661no—
154#127—1661no—
155#127C2×D8311yescharacteristic
156#124—1661no—
157#121—832no—
158#121—1661no—
159C4×D4—832no—
160C4×D4—1661no—
161C2×C42—831no—
162C4⋊C8—832no—
163#102—834no—
164#102C2×D8314yescomplemented
165C22⋊C8—832no—
166C4×C8—831no—
167C4×C8C2×D8311yescharacteristic, complemented
168C24—1662no—
169#41—3324no—
170C2×Q8—21no—
171C2×D4—21no—
172C2×D4—21no—
173C2×D4—3324no—
174C2×D4—3324no—
175C2×D4—3324no—
176C2×D4—832no—
177C2×D4—832no—
178C2×D4—1662no—
179C2×D4—1662no—
180C2×D4—1662no—
181C2×D4—1662no—
182C2×D4—1661no—
183C2×D4—1661no—
184C2×D4—1661no—
185C2×D4—1661no—
186C2×D4C22×D8312yes—
187C22×C4—831no—
188C22×C4—831no—
189C22×C4—831no—
190C22×C4—1662no—
191D8—3324no—
192D8—3324no—
193C2×C8—832no—
194C2×C8C22×D8312yes—
195C4⋊C4—21no—
196C4⋊C4C22×D8312yes—
197C22⋊C4—1662no—
198C22⋊C4—1662no—
199C42—831no—
200C42—831no—
201C42—831no—
202C42C22×D8311yescharacteristic
203C23—22no—
204C23—3324no—
205C23—3324no—
206C23—3324no—
207C23—831no—
208C23—1662no—
209C23—1662no—
210Q8—42no—
211D4—24no—
212D4—42no—
213D4—42no—
214D4—3324no—
215D4—3324no—
216D4—3324no—
217D4—3324no—
218D4—3322no—
219D4—3322no—
220D4—3322no—
221D4—3322no—
222D4—1664no—
223D4—1664no—
224C2×C4—22no—
225C2×C4—3324no—
226C2×C4—831no—
227C2×C4—834no—
228C2×C4—831no—
229C2×C4—831no—
230C2×C4—1662no—
231C2×C4—1662no—
232C2×C4D4×D8311yescharacteristic
233C2×C4D4×D8311yescharacteristic
234C2×C4C23×D8311yescharacteristic
235C8—22no—
236C8—1662no—
237C22—42no—
238C22—42no—
239C22—42no—
240C22—3322no—
241C22—3322no—
242C22—3322no—
243C22—3322no—
244C22—3322no—
245C22—3322no—
246C22—832no—
247C22—6648no—
248C22—1661no—
249C22—1661no—
250C22C2×D4×D8311yescharacteristic
251C4—42no—
252C4—832no—
253C4—1661no—
254C4—1661no—
255C4D8×D8312yes—
256C4#1458181912yes—
257C4#1458183812yes—
258C2—82no—
259C2—6642no—
260C2—6642no—
261C2—1662no—
262C2#1912826911yescharacteristic
263C2C2×D8×D8311yescharacteristic
264C2C2×#1458181911yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#486×C83C2
socleC22×C83C22×D4
hypercenter#486D83
hypocenterC83#3205
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C42 ◁ #486

Complements

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

NH ≅ G/Ntype
1. #19128252C2semidirect
2. #127×D83C2semidirect
3. #19128078C2semidirect
4. #19128077C2semidirect
5. #19128049C2semidirect
6. #19127895C2semidirect
7. #19127836C2semidirect
8. #19127752C2semidirect
9. #486×C83C2semidirect
14. C2×D664C22semidirect
15. C2×#13050575C22semidirect
17. #14581789C22semidirect
25. #14581754C22semidirect
27. Dic83⋊C8C22semidirect
28. #14581712C22semidirect
30. C4×C83⋊C8C22semidirect
31. C4⋊Q8×C83C22semidirect
33. #102×C83C22semidirect
34. C4×C8×C83C22semidirect
101. C4×D83D8semidirect
133. C83#3205semidirect
139. #486D83semidirect
149. C4⋊Q8C2×D83semidirect
164. #102C2×D83semidirect
167. C4×C8C2×D83semidirect

Maximal subgroups

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

subgroupindex
#191282522
#127×D832
#191280782
#191280772
#191280492
#191278952
#191278362
#191277522
#486×C832
#320583