Finite Group Databasebeta

← #24916708

Subgroups — #24916708

Order 10624.

758 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 758 conjugacy classes of subgroups — too many to draw. Only its 149 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
1C22×#13050590C211yesmaximal, characteristic, complemented
2C2×#14581825C211yesmaximal, characteristic, complemented
3C2×#14581820C211yesmaximal, characteristic, complemented
4#19128212C211yesmaximal, characteristic, complemented
5#19127865C214yesmaximal, complemented
6#19127848C214yesmaximal, complemented
7#19127711C211yesmaximal, characteristic, complemented
8#19127707C211yesmaximal, characteristic, complemented
9#407×C83C211yesmaximal, characteristic, complemented
10C2×#13050590—24no—
11C2×#13050590—24no—
12C2×#13050590—22no—
13C2×#13050590C2214yes—
14C2×#13050590C2211yescharacteristic
15C22×Dic166C2211yescharacteristic
16C22×D332C2211yescharacteristic
17C22×C22⋊D83—21no—
18C22×C4×D83—21no—
19#14581825—24no—
20#14581820—24no—
21C2×Dic332C2212yescomplemented
22C2×#13050579C2212yescomplemented
23C2×#13050579C2212yescomplemented
24C2×D664C2212yescomplemented
25#14581775—21no—
26#14581771C2212yescomplemented
27#14581766C2212yescomplemented
28#14581761—21no—
29C4×C22⋊D83—22no—
30#14581751C2211yescharacteristic
31#14581717C2214yescomplemented
32#14581714C2214yescomplemented
33C23×C4×C83C2211yescharacteristic
34C2×M16×C83C2211yescharacteristic
35C2×M16×C83C2211yescharacteristic
36C22⋊C8×C83C2214yescomplemented
37C23×D83—21no—
38#13050590—216no—
39#13050590—28no—
40#13050590—22no—
41#13050590—22no—
42#13050590—28no—
43C2×Dic166—22no—
44C2×Dic166—21no—
45C2×Dic166D414yescomplemented
46C2×Dic166C2311yescharacteristic
47C2×Dic166C2312yes—
48C2×D332—22no—
49C2×D332—21no—
50C2×D332D414yescomplemented
51C2×D332C2311yescharacteristic
52C2×D332C2312yes—
53C2×C22⋊D83—28no—
54C2×C22⋊D83—24no—
55C2×C22⋊D83—22no—
56C2×C22⋊D83—21no—
57C22×Dic83—21no—
58C2×C4×D83—28no—
59C2×C4×D83—24no—
60C2×C4×D83—21no—
61Dic332—24no—
62#13050579—24no—
63#13050579—24no—
64D664—24no—
65C4⋊Dic83C2312yes—
66Dic83⋊C4—22no—
67C22⋊Dic83—22no—
68C22⋊Dic83—21no—
69(C2×D83)⋊C4—22no—
70C4×Dic83—21no—
71C24×C83D411yescharacteristic
72C22×C4×C83—22no—
73C22×C4×C83—22no—
74C22×C4×C83D411yescharacteristic
75C22×C4×C83D412yes—
76C22×C4×C83C2312yes—
77C22×C4×C83C2311yescharacteristic
78M16×C83—22no—
79M16×C83—22no—
80C2×C8×C83C2312yes—
81C2×C8×C83C2312yes—
82C23×C83—22no—
83C23×C83—42no—
84C23×C83C2×D412yes—
85C23×C83C2×D411yescharacteristic
86C22×D83—24no—
87C22×D83—22no—
88C22×D83—21no—
89C22×D83—44no—
90C2×C4×C83—22no—
91C2×C4×C83—24no—
92C2×C4×C83—22no—
93C2×C4×C83—22no—
94C2×C4×C83C2×D412yes—
95C2×C4×C83C2×D412yes—
96C2×C4×C83C2×D412yes—
97C2×C4×C83C2×D411yescharacteristic
98C2×C4×C83C2411yescharacteristic
99C22⋊D83—216no—
100C22⋊D83—44no—
101C22⋊D83—42no—
102C2×Dic83—24no—
103C2×Dic83—22no—
104C2×Dic83—22no—
105C2×Dic83—21no—
106D332—24no—
107D332—22no—
108D332—21no—
109D332—21no—
110D332C2×D414yes—
111C4×D83—216no—
112C4×D83—42no—
113Dic166—24no—
114Dic166—22no—
115Dic166—21no—
116Dic166—21no—
117Dic166C2×D414yes—
118C8×C83—22no—
119C8×C83—22no—
120C22×C83—22no—
121C22×C83—22no—
122C22×C83—22no—
123C22×C83—22no—
124C22×C83—44no—
125C22×C83#12012yes—
126C22×C83C22×D411yescharacteristic
127C2×D83—24no—
128C2×D83—48no—
129C2×D83—44no—
130C2×D83—41no—
131C2×D83—42no—
132C4×C83—22no—
133C4×C83#12012yes—
134C4×C83C22×D411yescharacteristic
135C4×C83C22×D411yescharacteristic
136Dic83—24no—
137Dic83—41no—
138Dic83—42no—
139C2×C83—22no—
140C2×C83—42no—
141C2×C83C2×#12011yescharacteristic
142C2×C83#57512yes—
143D83—44no—
144D83—81no—
145#3005—831nomaximal, Sylow
146C83#300511yescharacteristic, complemented, Sylow
147C22×#41—831no—
148C2×#137—831no—
149C2×AGL(1,ℤ/8)—831no—
150#525—831no—
151#452—834no—
152#449—834no—
153#419—831no—
154#418—831no—
155#407D8311yescharacteristic, complemented
156C2×#41—831no—
157C2×#41—834no—
158C2×#41—1664no—
159C2×#41—1664no—
160C2×#41—1662no—
161C22×Q8—831no—
162C22×D4—831no—
163C22×D4—1661no—
164C23×C4—1661no—
165C23×C4C2×D8311yescharacteristic
166#137—1664no—
167AGL(1,ℤ/8)—1664no—
168C2×Q16—832no—
169C2×SD16—832no—
170C2×SD16—832no—
171C2×D8—832no—
172C2×M16C2×D8311yescharacteristic
173C2×M16C2×D8311yescharacteristic
174#123—1661no—
175C22⋊Q8—832no—
176#121—832no—
177#120—1661no—
178C4×D4—1662no—
179#117—831no—
180#103—834no—
181#102—834no—
182C22⋊C8C2×D8314yescomplemented
183C24—1661no—
184C24D33211yescharacteristic
185#41—1668no—
186#41—1668no—
187#41—16616no—
188#41—1662no—
189#41—1662no—
190C2×Q8—834no—
191C2×Q8—832no—
192C2×Q8—831no—
193C2×Q8—1662no—
194C2×Q8—1661no—
195C2×D4—834no—
196C2×D4—832no—
197C2×D4—831no—
198C2×D4—1662no—
199C2×D4—1661no—
200C2×D4—1668no—
201C2×D4—1664no—
202C2×D4—1662no—
203C2×D4—1661no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—1668no—
207C22×C4—1661no—
208C22×C4—1661no—
209C22×C4—1664no—
210C22×C4D33212yes—
211C22×C4D33211yescharacteristic
212C22×C4C22×D8312yes—
213C22×C4C22×D8311yescharacteristic
214Q16—1664no—
215SD16—1664no—
216SD16—1664no—
217D8—1664no—
218M16—22no—
219M16—22no—
220C2×C8C22×D8312yes—
221C2×C8C22×D8312yes—
222C4⋊C4—832no—
223C4⋊C4—1662no—
224C22⋊C4—1661no—
225C22⋊C4—1662no—
226C22⋊C4—1662no—
227C42—1661no—
228C23—22no—
229C23—42no—
230C23—3324no—
231C23—1661no—
232C23—1664no—
233C23—1662no—
234C23C2×D33212yes—
235C23C2×D33211yescharacteristic
236Q8—834no—
237Q8—1661no—
238Q8—1661no—
239Q8—1664no—
240Q8—1662no—
241D4—3324no—
242D4—3322no—
243D4—834no—
244D4—16616no—
245D4—1664no—
246D4—1661no—
247D4—1661no—
248D4—1662no—
249C2×C4—22no—
250C2×C4—24no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—3322no—
254C2×C4—16616no—
255C2×C4—1662no—
256C2×C4—1661no—
257C2×C4—1664no—
258C2×C4—1662no—
259C2×C4D4×D8312yes—
260C2×C4D4×D8312yes—
261C2×C4C2×D33212yes—
262C2×C4C2×D33211yescharacteristic
263C2×C4C23×D8311yescharacteristic
264C8—22no—
265C8—22no—
266C22—22no—
267C22—22no—
268C22—22no—
269C22—22no—
270C22—44no—
271C22—3328no—
272C22—3324no—
273C22—3321no—
274C22—3322no—
275C22—1664no—
276C22#1458176212yes—
277C22C22×D33211yescharacteristic
278C4—22no—
279C4—3321no—
280C4—3322no—
281C4—1664no—
282C4#1458176212yes—
283C4C2×D4×D8311yescharacteristic
284C4C2×D4×D8311yescharacteristic
285C2—22no—
286C2—42no—
287C2—3324no—
288C2—6641no—
289C2C2×#1458176211yescharacteristic
290C2#1912850112yes—
subgroupG/H
centerC22C22×D332
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#407×C83C2
socleC22×C83C22×D4
hypercenter#407D83
hypocenterC83#3005
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C23×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
1. C22×#13050590C2semidirect
2. C2×#14581825C2semidirect
3. C2×#14581820C2semidirect
4. #19128212C2semidirect
5. #19127865C2semidirect
6. #19127848C2semidirect
7. #19127711C2semidirect
8. #19127707C2semidirect
9. #407×C83C2semidirect
21. C2×Dic332C22semidirect
22. C2×#13050579C22semidirect
23. C2×#13050579C22semidirect
24. C2×D664C22semidirect
26. #14581771C22semidirect
27. #14581766C22semidirect
31. #14581717C22semidirect
32. #14581714C22semidirect
36. C22⋊C8×C83C22semidirect
45. C2×Dic166D4semidirect
50. C2×D332D4semidirect
146. C83#3005semidirect
155. #407D83semidirect
182. C22⋊C8C2×D83semidirect

Maximal subgroups

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

subgroupindex
C22×#130505902
C2×#145818252
C2×#145818202
#191282122
#191278652
#191278482
#191277112
#191277072
#407×C832
#300583