Finite Group Databasebeta

← #24919062

Subgroups — #24919062

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
3#19128336C211yesmaximal, characteristic, complemented
4#19127944C212yesmaximal, complemented
5#19127903C211yesmaximal, characteristic, complemented
6#19127893C211yesmaximal, characteristic, complemented
7#19127850C212yesmaximal, complemented
8#19127845C212yesmaximal, complemented
9C4×D664C211yesmaximal, characteristic, complemented
10#19127780C211yesmaximal, characteristic, complemented
11#492×C83C211yesmaximal, characteristic, complemented
12C2×#13050591C2212yescomplemented
13C2×D4×D83—21no—
14C2×D4×D83C2212yes—
15#14581819—28no—
16C2×#13050577C2212yes—
17C2×D664C2211yescharacteristic
18C2×#13050575C2212yescomplemented
19C2×#13050573C2212yescomplemented
20#14581777—22no—
21#14581768C2212yescomplemented
22#14581767—21no—
23#14581763—21no—
24#14581763C2212yes—
25#14581760—22no—
26C4×D332C2211yescharacteristic
27C4×D332C2211yescharacteristic
28C4×D332C2211yescharacteristic, complemented
29C4⋊C4×D83—21no—
30#14581726C2211yescharacteristic, complemented
31C4⋊(C83⋊C8)C2211yescharacteristic, complemented
32#14581716C2212yescomplemented
33#14581714C2211yescharacteristic
34#14581714C2211yescharacteristic, complemented
35#14581713C2212yescomplemented
36(C2×D83)⋊C8C2212yescomplemented
37C2×SD16×C83C2212yescomplemented
38C2×SD16×C83C2212yescomplemented
39C4⋊Q8×C83C2211yescharacteristic, complemented
40#127×C83C2211yescharacteristic
41C4×C8×C83C2211yescharacteristic, complemented
42C23×D83—22no—
43#13050591—24no—
44#13050591—42no—
45C2×D332—22no—
46C2×D332D411yescharacteristic
47C2×D332C2311yescharacteristic
48C2×D332C2311yescharacteristic
49C2×C22⋊D83—22no—
50C2×C22⋊D83—22no—
51D4×D83—24no—
52D4×D83—42no—
53D4×D83—44no—
54C2×C4×D83—21no—
55C2×C4×D83—21no—
56C2×C4×D83—22no—
57C2×C4×D83C2312yes—
58#13050577—24no—
59D664D414yescomplemented
60#13050575—24no—
61#13050573—24no—
62C2×C83⋊C8C2312yes—
63C4⋊Dic83D411yescharacteristic
64C4⋊Dic83C2311yescharacteristic
65C4⋊Dic83C2311yescharacteristic
66Dic83⋊C4—22no—
67C22⋊Dic83—22no—
68(C2×D83)⋊C4—22no—
69(C2×D83)⋊C4—21no—
70(C2×D83)⋊C4—21no—
71(C2×D83)⋊C4D412yes—
72C2×Q8×C83C2312yes—
73C2×D4×C83—21no—
74C2×D4×C83—21no—
75C2×D4×C83C2312yes—
76SD16×C83—24no—
77SD16×C83—42no—
78C2×C8×C83C2311yescharacteristic
79C2×C8×C83C2311yescharacteristic
80C4⋊C4×C83—21no—
81C4⋊C4×C83—21no—
82C42×C83C2311yescharacteristic
83C23×C83—22no—
84C22×D83—21no—
85C22×D83—21no—
86C22×D83—44no—
87C22×D83—44no—
88C22×D83—82no—
89C22×D83C2×D412yes—
90Q8×C83—22no—
91Q8×C83—22no—
92D4×C83—22no—
93D4×C83—22no—
94D4×C83—42no—
95D4×C83—81no—
96C2×C4×C83—22no—
97C2×C4×C83C2×D411yescharacteristic
98C2×C4×C83C2×D411yescharacteristic
99C2×C4×C83C2411yescharacteristic
100C22⋊D83—44no—
101C22⋊D83—44no—
102C2×Dic83—21no—
103C2×Dic83—21no—
104C2×Dic83C2×D412yes—
105D332—22no—
106D332—21no—
107D332—21no—
108D332—44no—
109D332C2×D412yes—
110D332C2×D412yescomplemented
111C4×D83—22no—
112C4×D83—22no—
113C4×D83—44no—
114C4×D83—42no—
115C4×D83—42no—
116C8×C83—21no—
117C8×C83C2×D412yescomplemented
118C83⋊C8—22no—
119C22×C83—42no—
120C22×C83—42no—
121C22×C83—42no—
122C22×C83C22×D411yescharacteristic
123C2×D83—22no—
124C2×D83—22no—
125C2×D83—22no—
126C2×D83—22no—
127C2×D83—22no—
128C2×D83—41no—
129C2×D83—41no—
130C2×D83—41no—
131C2×D83—41no—
132C2×D83—88no—
133C4×C83—21no—
134C4×C83—42no—
135C4×C83AGL(1,ℤ/8)12yescomplemented
136C4×C83C22×D411yescharacteristic
137C4×C83H(5,𝔽2)11yescharacteristic
138Dic83—22no—
139Dic83—41no—
140Dic83—41no—
141C2×C83—82no—
142C2×C83D4211yescharacteristic
143C2×C83C2×AGL(1,ℤ/8)11yescharacteristic
144C2×C83#57711yescharacteristic
145D83—42no—
146D83—42no—
147D83—42no—
148#3272—831nomaximal, Sylow
149C83#327211yescharacteristic, complemented, Sylow
150C2×AGL(1,ℤ/8)—832no—
151#547—831no—
152D42—831no—
153#492D8311yescharacteristic, complemented
154#468—832no—
155#461—831no—
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×SD16C2×D8312yescomplemented
167C2×SD16C2×D8312yescomplemented
168C2×D8—832no—
169C2×D8—831no—
170C2×M16—832no—
171C4⋊Q8C2×D8311yescharacteristic, complemented
172#127C2×D8311yescharacteristic
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×D4—831no—
182C2×C4⋊C4—1661no—
183#107—831no—
184C4⋊C8—831no—
185#103—832no—
186#102—832no—
187#102—831no—
188#102—831no—
189C22⋊C8—832no—
190C4×C8C2×D8311yescharacteristic, complemented
191C24—1662no—
192#41—3322no—
193#41—1664no—
194C2×Q8C22×D8312yes—
195C2×D4—21no—
196C2×D4—21no—
197C2×D4—3324no—
198C2×D4—3322no—
199C2×D4—831no—
200C2×D4—831no—
201C2×D4—831no—
202C2×D4—1662no—
203C2×D4—1662no—
204C2×D4—1664no—
205C2×D4—1662no—
206C2×D4C22×D8312yes—
207C22×C4—832no—
208C22×C4—1662no—
209C22×C4—1661no—
210C22×C4—1661no—
211SD16—24no—
212SD16—42no—
213SD16—1664no—
214D8—834no—
215D8—1664no—
216M16—1664no—
217C2×C8—832no—
218C2×C8C22×D8311yescharacteristic
219C2×C8C22×D8311yescharacteristic
220C4⋊C4—21no—
221C4⋊C4—21no—
222C4⋊C4—831no—
223C4⋊C4—831no—
224C4⋊C4—831no—
225C4⋊C4—1662no—
226C22⋊C4—832no—
227C22⋊C4—1662no—
228C22⋊C4—1662no—
229C22⋊C4—1661no—
230C22⋊C4—1661no—
231C42C22×D8311yescharacteristic
232C23—22no—
233C23—3324no—
234C23—3324no—
235C23—832no—
236C23—6642no—
237C23—1661no—
238C23—1661no—
239Q8—22no—
240Q8—22no—
241D4—22no—
242D4—22no—
243D4—42no—
244D4—81no—
245D4—3324no—
246D4—3324no—
247D4—3324no—
248D4—832no—
249D4—832no—
250D4—1662no—
251D4—1661no—
252D4—1661no—
253C2×C4—22no—
254C2×C4—3324no—
255C2×C4—3322no—
256C2×C4—3322no—
257C2×C4—832no—
258C2×C4—1661no—
259C2×C4—1661no—
260C2×C4—1662no—
261C2×C4—1662no—
262C2×C4D4×D8311yescharacteristic
263C2×C4D4×D8311yescharacteristic
264C2×C4C23×D8311yescharacteristic
265C8—21no—
266C8—1662no—
267C8D4×D8312yescomplemented
268C22—42no—
269C22—42no—
270C22—42no—
271C22—3321no—
272C22—3321no—
273C22—3321no—
274C22—3321no—
275C22—6648no—
276C22—1662no—
277C22—1662no—
278C22—1662no—
279C22—1662no—
280C22—1662no—
281C22C2×D4×D8311yescharacteristic
282C4—21no—
283C4—42no—
284C4—3321no—
285C4—3321no—
286C4—1662no—
287C4#1458181912yescomplemented
288C4C2×D4×D8311yescharacteristic
289C4#1458183711yescharacteristic
290C2—82no—
291C2—3322no—
292C2—3322no—
293C2—3322no—
294C2#1912833611yescharacteristic
295C2C2×#1458181911yescharacteristic
296C2#1912850711yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#492×C83C2
socleC22×C83C22×D4
hypercenter#492D83
hypocenterC83#3272
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C42 ◁ #492

Complements

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

NH ≅ G/Ntype
1. C2×#14581819C2semidirect
2. #19128354C2semidirect
3. #19128336C2semidirect
4. #19127944C2semidirect
5. #19127903C2semidirect
6. #19127893C2semidirect
7. #19127850C2semidirect
8. #19127845C2semidirect
9. C4×D664C2semidirect
10. #19127780C2semidirect
11. #492×C83C2semidirect
12. C2×#13050591C22semidirect
18. C2×#13050575C22semidirect
19. C2×#13050573C22semidirect
21. #14581768C22semidirect
28. C4×D332C22semidirect
30. #14581726C22semidirect
31. C4⋊(C83⋊C8)C22semidirect
32. #14581716C22semidirect
34. #14581714C22semidirect
35. #14581713C22semidirect
36. (C2×D83)⋊C8C22semidirect
37. C2×SD16×C83C22semidirect
38. C2×SD16×C83C22semidirect
39. C4⋊Q8×C83C22semidirect
41. C4×C8×C83C22semidirect
59. D664D4semidirect
110. D332C2×D4semidirect
117. C8×C83C2×D4semidirect
135. C4×C83AGL(1,ℤ/8)semidirect
149. C83#3272semidirect
153. #492D83semidirect
166. C2×SD16C2×D83semidirect
167. C2×SD16C2×D83semidirect
171. C4⋊Q8C2×D83semidirect
190. C4×C8C2×D83semidirect
267. C8D4×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
#191283362
#191279442
#191279032
#191278932
#191278502
#191278452
C4×D6642
#191277802
#492×C832
#327283