Finite Group Databasebeta

← #24919066

Subgroups — #24919066

Order 10624.

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

normal subgroups only

This group has 546 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
1C2×#14581819C212yesmaximal, complemented
2#19128354C211yesmaximal, characteristic, complemented
3D4×D332C211yesmaximal, characteristic, complemented
4#19128075C211yesmaximal, characteristic, complemented
5#19127950C212yesmaximal, complemented
6#19127897C211yesmaximal, characteristic, complemented
7#19127850C212yesmaximal, complemented
8#19127849C212yesmaximal, complemented
9C4×#13050575C211yesmaximal, characteristic, complemented
10#19127778C211yesmaximal, characteristic, complemented
11#461×C83C211yesmaximal, characteristic, complemented
12C2×#13050591C2212yescomplemented
13C22×D332—21no—
14C2×D4×D83C2212yes—
15#14581819—28no—
16C2×#13050577C2212yescomplemented
17C2×#13050577C2212yescomplemented
18C2×D664C2212yescomplemented
19C2×#13050575C2211yescharacteristic
20C2×#13050573C2212yescomplemented
21#14581789C2211yescharacteristic
22#14581777—22no—
23#14581768C2212yescomplemented
24#14581767—21no—
25#14581767C2212yes—
26#14581766—21no—
27#14581762—22no—
28C4×D332C2211yescharacteristic
29C4×D332C2211yescharacteristic, complemented
30C4⋊C4×D83—21no—
31#14581725C2211yescharacteristic, complemented
32#14581714C2212yescomplemented
33#14581713C2211yescharacteristic
34#14581712C2211yescharacteristic, complemented
35(C2×D83)⋊C8C2212yescomplemented
36C4×C83⋊C8C2211yescharacteristic, complemented
37C2×SD16×C83C2212yes—
38C4⋊Q8×C83C2211yescharacteristic, complemented
39C4×D4×C83C2211yescharacteristic
40C4⋊C8×C83C2211yescharacteristic, complemented
41#103×C83C2212yescomplemented
42C23×D83—22no—
43#13050591—24no—
44#13050591—42no—
45C2×D332—22no—
46C2×D332—22no—
47C2×D332—21no—
48C2×D332—21no—
49C2×D332—44no—
50C2×D332D411yescharacteristic
51C2×D332C2312yes—
52C2×D332C2311yescharacteristic
53C2×C22⋊D83—22no—
54D4×D83—24no—
55D4×D83—42no—
56C2×C4×D83—21no—
57C2×C4×D83—22no—
58C2×C4×D83C2312yes—
59#13050577—24no—
60#13050577—42no—
61D664—24no—
62#13050575D414yescomplemented
63#13050573—24no—
64C2×C83⋊C8C2311yescharacteristic
65C2×C83⋊C8C2311yescharacteristic
66C4⋊Dic83D411yescharacteristic
67C4⋊Dic83C2311yescharacteristic
68Dic83⋊C4—22no—
69(C2×D83)⋊C4—22no—
70(C2×D83)⋊C4—21no—
71(C2×D83)⋊C4—22no—
72(C2×D83)⋊C4D412yes—
73C2×Q8×C83C2312yes—
74C2×D4×C83C2311yescharacteristic
75C22×C4×C83—21no—
76SD16×C83—24no—
77C2×C8×C83C2312yes—
78C4⋊C4×C83—21no—
79C4⋊C4×C83—21no—
80C4⋊C4×C83C2311yescharacteristic
81C22⋊C4×C83—21no—
82C42×C83C2311yescharacteristic
83C23×C83—21no—
84C22×D83—21no—
85C22×D83—22no—
86C22×D83—44no—
87C22×D83—44no—
88C22×D83—82no—
89C22×D83C2×D412yes—
90Q8×C83—22no—
91Q8×C83—22no—
92D4×C83—21no—
93D4×C83C2×D412yes—
94C2×C4×C83—21no—
95C2×C4×C83—22no—
96C2×C4×C83—42no—
97C2×C4×C83C2×D411yescharacteristic
98C2×C4×C83C2×D411yescharacteristic
99C2×C4×C83C2411yescharacteristic
100C22⋊D83—44no—
101C2×Dic83—21no—
102C2×Dic83C2×D412yes—
103D332—22no—
104D332—22no—
105D332—21no—
106D332—22no—
107D332—44no—
108D332—44no—
109D332—42no—
110D332—81no—
111D332C2×D412yescomplemented
112C4×D83—22no—
113C4×D83—22no—
114C4×D83—44no—
115C4×D83—42no—
116C8×C83—22no—
117C83⋊C8—21no—
118C83⋊C8C2×D412yescomplemented
119C22×C83—22no—
120C22×C83—41no—
121C22×C83—41no—
122C22×C83C22×D411yescharacteristic
123C2×D83—22no—
124C2×D83—22no—
125C2×D83—22no—
126C2×D83—22no—
127C2×D83—41no—
128C2×D83—41no—
129C2×D83—42no—
130C2×D83—42no—
131C2×D83—42no—
132C2×D83—88no—
133C4×C83—21no—
134C4×C83—42no—
135C4×C83—41no—
136C4×C83AGL(1,ℤ/8)12yescomplemented
137C4×C83C22×D411yescharacteristic
138C4×C83H(5,𝔽2)11yescharacteristic
139Dic83—22no—
140Dic83—41no—
141C2×C83—42no—
142C2×C83D4211yescharacteristic
143C2×C83C2×AGL(1,ℤ/8)11yescharacteristic
144C2×C83#57711yescharacteristic
145D83—42no—
146D83—42no—
147D83—82no—
148#3272—831nomaximal, Sylow
149C83#327211yescharacteristic, complemented, Sylow
150C2×AGL(1,ℤ/8)—832no—
151#547—831no—
152D42—831no—
153#492—831no—
154#468—832no—
155#461D8311yescharacteristic, complemented
156#459—831no—
157#450—832no—
158#449—832no—
159C4×D8—831no—
160#436—831no—
161C2×#41—832no—
162C22×D4—832no—
163C22×D4—1661no—
164AGL(1,ℤ/8)—1668no—
165C2×SD16—832no—
166C2×SD16—832no—
167C2×SD16C2×D8312yes—
168C2×D8—831no—
169C2×D8—832no—
170C2×M16—832no—
171C4⋊Q8C2×D8311yescharacteristic, complemented
172#127—831no—
173#123—1662no—
174C22⋊Q8—832no—
175#121—832no—
176#121—1661no—
177#121—1661no—
178#120—1662no—
179C4×D4—831no—
180C4×D4—831no—
181C4×D4C2×D8311yescharacteristic
182C2×C4⋊C4—1661no—
183#107—831no—
184C4⋊C8C2×D8311yescharacteristic, complemented
185#103C2×D8312yescomplemented
186#102—831no—
187#102—832no—
188#102—831no—
189C22⋊C8—832no—
190C4×C8—831no—
191C24—1662no—
192#41—3322no—
193#41—1664no—
194C2×Q8C22×D8312yes—
195C2×D4—3324no—
196C2×D4—3322no—
197C2×D4—831no—
198C2×D4—831no—
199C2×D4—832no—
200C2×D4—1662no—
201C2×D4—1662no—
202C2×D4—1662no—
203C2×D4—1664no—
204C2×D4—1661no—
205C2×D4—1661no—
206C2×D4C22×D8311yescharacteristic
207C22×C4—21no—
208C22×C4—832no—
209C22×C4—1662no—
210C22×C4—1661no—
211SD16—24no—
212SD16—3322no—
213SD16—1664no—
214D8—834no—
215D8—1664no—
216M16—1664no—
217C2×C8—831no—
218C2×C8—831no—
219C2×C8C22×D8312yes—
220C4⋊C4—21no—
221C4⋊C4—21no—
222C4⋊C4—831no—
223C4⋊C4—831no—
224C4⋊C4—1662no—
225C4⋊C4C22×D8311yescharacteristic
226C22⋊C4—21no—
227C22⋊C4—832no—
228C22⋊C4—1662no—
229C22⋊C4—1662no—
230C22⋊C4—1661no—
231C42C22×D8311yescharacteristic
232C23—21no—
233C23—3324no—
234C23—3324no—
235C23—832no—
236C23—6642no—
237C23—1662no—
238C23—1661no—
239Q8—22no—
240Q8—22no—
241D4—21no—
242D4—3324no—
243D4—3324no—
244D4—3324no—
245D4—3322no—
246D4—832no—
247D4—6641no—
248D4—1662no—
249D4—1662no—
250D4—1662no—
251D4—1661no—
252D4D4×D8312yes—
253C2×C4—21no—
254C2×C4—22no—
255C2×C4—42no—
256C2×C4—3324no—
257C2×C4—3322no—
258C2×C4—832no—
259C2×C4—1661no—
260C2×C4—1662no—
261C2×C4—1662no—
262C2×C4D4×D8311yescharacteristic
263C2×C4D4×D8311yescharacteristic
264C2×C4C23×D8311yescharacteristic
265C8—22no—
266C8—832no—
267C8—1661no—
268C22—22no—
269C22—41no—
270C22—41no—
271C22—3321no—
272C22—3321no—
273C22—3322no—
274C22—3322no—
275C22—3322no—
276C22—6648no—
277C22—1662no—
278C22—1662no—
279C22—1662no—
280C22—1662no—
281C22C2×D4×D8311yescharacteristic
282C4—21no—
283C4—42no—
284C4—41no—
285C4—3321no—
286C4—1662no—
287C4#1458181912yescomplemented
288C4C2×D4×D8311yescharacteristic
289C4#1458183811yescharacteristic
290C2—42no—
291C2—3322no—
292C2—3322no—
293C2—6642no—
294C2#1912833511yescharacteristic
295C2C2×#1458181911yescharacteristic
296C2#1912851011yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#461×C83C2
socleC22×C83C22×D4
hypercenter#461D83
hypocenterC83#3272
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
1. C2×#14581819C2semidirect
2. #19128354C2semidirect
3. D4×D332C2semidirect
4. #19128075C2semidirect
5. #19127950C2semidirect
6. #19127897C2semidirect
7. #19127850C2semidirect
8. #19127849C2semidirect
9. C4×#13050575C2semidirect
10. #19127778C2semidirect
11. #461×C83C2semidirect
12. C2×#13050591C22semidirect
16. C2×#13050577C22semidirect
17. C2×#13050577C22semidirect
18. C2×D664C22semidirect
20. C2×#13050573C22semidirect
23. #14581768C22semidirect
29. C4×D332C22semidirect
31. #14581725C22semidirect
32. #14581714C22semidirect
34. #14581712C22semidirect
35. (C2×D83)⋊C8C22semidirect
36. C4×C83⋊C8C22semidirect
38. C4⋊Q8×C83C22semidirect
40. C4⋊C8×C83C22semidirect
41. #103×C83C22semidirect
62. #13050575D4semidirect
111. D332C2×D4semidirect
118. C83⋊C8C2×D4semidirect
136. C4×C83AGL(1,ℤ/8)semidirect
149. C83#3272semidirect
155. #461D83semidirect
171. C4⋊Q8C2×D83semidirect
184. C4⋊C8C2×D83semidirect
185. #103C2×D83semidirect
287. C4#14581819semidirect

Maximal subgroups

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

subgroupindex
C2×#145818192
#191283542
D4×D3322
#191280752
#191279502
#191278972
#191278502
#191278492
C4×#130505752
#191277782
#461×C832
#327283