Finite Group Databasebeta

← #38754315

Subgroups — #38754315

Order 25216.

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×#20263353C212yesmaximal, complemented
2#27161732C211yesmaximal, characteristic, complemented
3D4×D788C211yesmaximal, characteristic, complemented
4#27161453C211yesmaximal, characteristic, complemented
5#27161328C212yesmaximal, complemented
6#27161275C211yesmaximal, characteristic, complemented
7#27161228C212yesmaximal, complemented
8#27161227C212yesmaximal, complemented
9C4×#15691343C211yesmaximal, characteristic, complemented
10#27161156C211yesmaximal, characteristic, complemented
11#461×C197C211yesmaximal, characteristic, complemented
12C2×#15691359C2212yescomplemented
13C22×D788—21no—
14C2×D4×D197C2212yes—
15#20263353—28no—
16C2×#15691345C2212yescomplemented
17C2×#15691345C2212yescomplemented
18C2×D1576C2212yescomplemented
19C2×#15691343C2211yescharacteristic
20C2×#15691341C2212yescomplemented
21#20263323C2211yescharacteristic
22#20263311—22no—
23#20263302C2212yescomplemented
24#20263301—21no—
25#20263301C2212yes—
26#20263300—21no—
27#20263296—22no—
28C4×D788C2211yescharacteristic
29C4×D788C2211yescharacteristic, complemented
30C4⋊C4×D197—21no—
31#20263259C2211yescharacteristic, complemented
32#20263248C2212yescomplemented
33#20263247C2211yescharacteristic
34#20263246C2211yescharacteristic, complemented
35#20263238C2212yescomplemented
36C4×#13096802C2211yescharacteristic, complemented
37C2×SD16×C197C2212yes—
38C4⋊Q8×C197C2211yescharacteristic, complemented
39C4×D4×C197C2211yescharacteristic
40C4⋊C8×C197C2211yescharacteristic, complemented
41#103×C197C2212yescomplemented
42C23×D197—22no—
43#15691359—24no—
44#15691359—42no—
45C2×D788—22no—
46C2×D788—22no—
47C2×D788—21no—
48C2×D788—21no—
49C2×D788—44no—
50C2×D788D411yescharacteristic
51C2×D788C2312yes—
52C2×D788C2311yescharacteristic
53C2×C22⋊D197—22no—
54D4×D197—24no—
55D4×D197—42no—
56C2×C4×D197—22no—
57C2×C4×D197—21no—
58C2×C4×D197C2312yes—
59#15691345—24no—
60#15691345—42no—
61D1576—24no—
62#15691343D414yescomplemented
63#15691341—24no—
64C2×#13096802C2311yescharacteristic
65C2×#13096802C2311yescharacteristic
66C4⋊Dic197D411yescharacteristic
67C4⋊Dic197C2311yescharacteristic
68#15691337—22no—
69#15691335—22no—
70#15691335—22no—
71#15691335—21no—
72#15691335D412yes—
73C2×Q8×C197C2312yes—
74C2×D4×C197C2311yescharacteristic
75C22×C4×C197—21no—
76SD16×C197—24no—
77C2×C8×C197C2312yes—
78C4⋊C4×C197—21no—
79C4⋊C4×C197—21no—
80C4⋊C4×C197C2311yescharacteristic
81C22⋊C4×C197—21no—
82C42×C197C2311yescharacteristic
83C22×D197—21no—
84C22×D197—22no—
85C22×D197—44no—
86C22×D197—44no—
87C22×D197—82no—
88C22×D197C2×D412yes—
89D788—22no—
90D788—22no—
91D788—21no—
92D788—22no—
93D788—44no—
94D788—44no—
95D788—42no—
96D788—81no—
97D788C2×D412yescomplemented
98C22⋊D197—44no—
99C2×Dic197—21no—
100C2×Dic197C2×D412yes—
101C4×D197—22no—
102C4×D197—22no—
103C4×D197—44no—
104C4×D197—42no—
105#13096802—21no—
106#13096802C2×D412yescomplemented
107C23×C197—21no—
108Q8×C197—22no—
109Q8×C197—22no—
110D4×C197—21no—
111D4×C197C2×D412yes—
112C2×C4×C197—21no—
113C2×C4×C197—22no—
114C2×C4×C197—42no—
115C2×C4×C197C2×D411yescharacteristic
116C2×C4×C197C2×D411yescharacteristic
117C2×C4×C197C2411yescharacteristic
118C8×C197—22no—
119C22×C197—22no—
120C22×C197—41no—
121C22×C197—41no—
122C22×C197C22×D411yescharacteristic
123C2×D197—22no—
124C2×D197—22no—
125C2×D197—22no—
126C2×D197—22no—
127C2×D197—41no—
128C2×D197—41no—
129C2×D197—42no—
130C2×D197—42no—
131C2×D197—42no—
132C2×D197—88no—
133C4×C197—21no—
134C4×C197—42no—
135C4×C197—41no—
136C4×C197AGL(1,ℤ/8)12yescomplemented
137C4×C197C22×D411yescharacteristic
138C4×C197H(5,𝔽2)11yescharacteristic
139Dic197—22no—
140Dic197—41no—
141C2×C197—42no—
142C2×C197D4211yescharacteristic
143C2×C197C2×AGL(1,ℤ/8)11yescharacteristic
144C2×C197#57711yescharacteristic
145D197—42no—
146D197—42no—
147D197—82no—
148C197#327211yescharacteristic, complemented, Sylow
149#3272—1971nomaximal, Sylow
150C2×AGL(1,ℤ/8)—1972no—
151#547—1971no—
152D42—1971no—
153#492—1971no—
154#468—1972no—
155#461D19711yescharacteristic, complemented
156#459—1971no—
157#450—1972no—
158#449—1972no—
159C4×D8—1971no—
160#436—1971no—
161C2×#41—1972no—
162C22×D4—3941no—
163C22×D4—1972no—
164AGL(1,ℤ/8)—3948no—
165C2×SD16—1972no—
166C2×SD16—1972no—
167C2×SD16C2×D19712yes—
168C2×D8—1971no—
169C2×D8—1972no—
170C2×M16—1972no—
171C4⋊Q8C2×D19711yescharacteristic, complemented
172#127—1971no—
173#123—3942no—
174C22⋊Q8—1972no—
175#121—3941no—
176#121—3941no—
177#121—1972no—
178#120—3942no—
179C4×D4—1971no—
180C4×D4—1971no—
181C4×D4C2×D19711yescharacteristic
182C2×C4⋊C4—3941no—
183#107—1971no—
184C4⋊C8C2×D19711yescharacteristic, complemented
185#103C2×D19712yescomplemented
186#102—1971no—
187#102—1972no—
188#102—1971no—
189C22⋊C8—1972no—
190C4×C8—1971no—
191C24—3942no—
192#41—7882no—
193#41—3944no—
194C2×Q8C22×D19712yes—
195C2×D4—7884no—
196C2×D4—7882no—
197C2×D4—3942no—
198C2×D4—3942no—
199C2×D4—3944no—
200C2×D4—3942no—
201C2×D4—3941no—
202C2×D4—3941no—
203C2×D4—1971no—
204C2×D4—1972no—
205C2×D4—1971no—
206C2×D4C22×D19711yescharacteristic
207C22×C4—21no—
208C22×C4—3942no—
209C22×C4—3941no—
210C22×C4—1972no—
211SD16—24no—
212SD16—7882no—
213SD16—3944no—
214D8—3944no—
215D8—1974no—
216M16—3944no—
217C2×C8—1971no—
218C2×C8—1971no—
219C2×C8C22×D19712yes—
220C4⋊C4—21no—
221C4⋊C4—21no—
222C4⋊C4—3942no—
223C4⋊C4—1971no—
224C4⋊C4—1971no—
225C4⋊C4C22×D19711yescharacteristic
226C22⋊C4—21no—
227C22⋊C4—3942no—
228C22⋊C4—3942no—
229C22⋊C4—3941no—
230C22⋊C4—1972no—
231C42C22×D19711yescharacteristic
232C23—21no—
233C23—7884no—
234C23—7884no—
235C23—15762no—
236C23—3942no—
237C23—3941no—
238C23—1972no—
239Q8—22no—
240Q8—22no—
241D4—21no—
242D4—7884no—
243D4—7884no—
244D4—7884no—
245D4—7882no—
246D4—15761no—
247D4—3942no—
248D4—3942no—
249D4—3942no—
250D4—3941no—
251D4—1972no—
252D4D4×D19712yes—
253C2×C4—21no—
254C2×C4—22no—
255C2×C4—42no—
256C2×C4—7884no—
257C2×C4—7882no—
258C2×C4—3941no—
259C2×C4—3942no—
260C2×C4—3942no—
261C2×C4—1972no—
262C2×C4D4×D19711yescharacteristic
263C2×C4D4×D19711yescharacteristic
264C2×C4C23×D19711yescharacteristic
265C8—22no—
266C8—3941no—
267C8—1972no—
268C22—22no—
269C22—41no—
270C22—41no—
271C22—7881no—
272C22—7881no—
273C22—7882no—
274C22—7882no—
275C22—7882no—
276C22—15768no—
277C22—3942no—
278C22—3942no—
279C22—3942no—
280C22—3942no—
281C22C2×D4×D19711yescharacteristic
282C4—21no—
283C4—41no—
284C4—42no—
285C4—7881no—
286C4—3942no—
287C4#2026335312yescomplemented
288C4C2×D4×D19711yescharacteristic
289C4#2026337211yescharacteristic
290C2—42no—
291C2—7882no—
292C2—7882no—
293C2—15762no—
294C2#2716171311yescharacteristic
295C2C2×#2026335311yescharacteristic
296C2#2716188811yescharacteristic
subgroupG/H
centerC22C2×D4×D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#461×C197C2
socleC22×C197C22×D4
hypercenter#461D197
hypocenterC197#3272
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
1. C2×#20263353C2semidirect
2. #27161732C2semidirect
3. D4×D788C2semidirect
4. #27161453C2semidirect
5. #27161328C2semidirect
6. #27161275C2semidirect
7. #27161228C2semidirect
8. #27161227C2semidirect
9. C4×#15691343C2semidirect
10. #27161156C2semidirect
11. #461×C197C2semidirect
12. C2×#15691359C22semidirect
16. C2×#15691345C22semidirect
17. C2×#15691345C22semidirect
18. C2×D1576C22semidirect
20. C2×#15691341C22semidirect
23. #20263302C22semidirect
29. C4×D788C22semidirect
31. #20263259C22semidirect
32. #20263248C22semidirect
34. #20263246C22semidirect
35. #20263238C22semidirect
36. C4×#13096802C22semidirect
38. C4⋊Q8×C197C22semidirect
40. C4⋊C8×C197C22semidirect
41. #103×C197C22semidirect
62. #15691343D4semidirect
97. D788C2×D4semidirect
106. #13096802C2×D4semidirect
136. C4×C197AGL(1,ℤ/8)semidirect
148. C197#3272semidirect
155. #461D197semidirect
171. C4⋊Q8C2×D197semidirect
184. C4⋊C8C2×D197semidirect
185. #103C2×D197semidirect
287. C4#20263353semidirect

Maximal subgroups

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

subgroupindex
C2×#202633532
#271617322
D4×D7882
#271614532
#271613282
#271612752
#271612282
#271612272
C4×#156913432
#271611562
#461×C1972
#3272197