Finite Group Databasebeta

← #38754313

Subgroups — #38754313

Order 25216.

546 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 546 conjugacy classes of subgroups — too many to draw. Only its 133 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×#20263350C212yesmaximal, complemented
2#27161733C211yesmaximal, characteristic, complemented
3D4×D788C211yesmaximal, characteristic, complemented
4#27161452C211yesmaximal, characteristic, complemented
5#27161326C212yesmaximal, complemented
6#27161284C211yesmaximal, characteristic, complemented
7#27161274C211yesmaximal, characteristic, complemented
8#27161231C212yesmaximal, complemented
9#27161225C212yesmaximal, complemented
10#27161158C211yesmaximal, characteristic, complemented
11C4×D8×C197C211yesmaximal, characteristic, complemented
12C2×#15691357C2212yescomplemented
13C22×D788—21no—
14C2×D4×D197C2212yes—
15#20263350—28no—
16C2×#15691347C2212yescomplemented
17C2×#15691347C2212yescomplemented
18C2×#15691346C2212yes—
19C2×#15691343C2212yescomplemented
20C2×#15691341C2212yescomplemented
21#20263326C2211yescharacteristic, complemented
22#20263323C2211yescharacteristic
23#20263310—22no—
24#20263306C2212yescomplemented
25#20263301C2212yes—
26#20263300—21no—
27#20263300—21no—
28#20263296—22no—
29C4×D788C2211yescharacteristic
30C2×C4⋊Dic197—21no—
31C4⋊#13096802C2211yescharacteristic, complemented
32#20263249C2212yescomplemented
33#20263246C2212yescomplemented
34#20263238C2212yescomplemented
35C2×D8×C197C2211yescharacteristic
36C4×D4×C197C2211yescharacteristic
37C4×D4×C197C2211yescharacteristic, complemented
38#107×C197C2211yescharacteristic, complemented
39#102×C197C2211yescharacteristic
40#102×C197C2211yescharacteristic, complemented
41C4×C8×C197C2211yescharacteristic, complemented
42C23×D197—22no—
43#15691357—24no—
44#15691357—42no—
45C2×Dic394C2312yes—
46C2×D788—22no—
47C2×D788—21no—
48C2×D788—21no—
49C2×D788—44no—
50C2×D788D411yescharacteristic
51C2×D788C2312yes—
52C2×C22⋊D197—22no—
53C2×C22⋊D197—22no—
54D4×D197—24no—
55D4×D197—42no—
56C22×Dic197—22no—
57C2×C4×D197C2312yes—
58#15691347—24no—
59#15691347—42no—
60#15691346—24no—
61#15691343—24no—
62#15691341—24no—
63C2×#13096802C2312yes—
64C4⋊Dic197—22no—
65C4⋊Dic197—21no—
66C4⋊Dic197—21no—
67C4⋊Dic197D411yescharacteristic
68#15691335—22no—
69#15691335—22no—
70#15691335D412yes—
71C2×D4×C197C2311yescharacteristic
72C2×D4×C197C2311yescharacteristic
73C22×C4×C197—21no—
74C22×C4×C197—21no—
75D8×C197D414yescomplemented
76C2×C8×C197C2311yescharacteristic
77C2×C8×C197C2311yescharacteristic
78C4⋊C4×C197C2311yescharacteristic
79C4⋊C4×C197C2311yescharacteristic
80C22⋊C4×C197—21no—
81C22⋊C4×C197—21no—
82C42×C197C2311yescharacteristic
83C22×D197—22no—
84C22×D197—44no—
85C22×D197—44no—
86C22×D197—82no—
87C22×D197C2×D412yes—
88Dic394—22no—
89Dic394—22no—
90D788—22no—
91D788—22no—
92D788—22no—
93D788—44no—
94D788—42no—
95D788—81no—
96C22⋊D197—44no—
97C22⋊D197—44no—
98C2×Dic197—22no—
99C2×Dic197—44no—
100C2×Dic197C2×D412yes—
101C4×D197—22no—
102C4×D197—22no—
103#13096802—22no—
104C23×C197—21no—
105C23×C197—21no—
106D4×C197—21no—
107D4×C197—21no—
108D4×C197C2×D412yes—
109D4×C197C2×D412yescomplemented
110C2×C4×C197—21no—
111C2×C4×C197—21no—
112C2×C4×C197—42no—
113C2×C4×C197—42no—
114C2×C4×C197C2×D411yescharacteristic
115C2×C4×C197C2×D411yescharacteristic
116C2×C4×C197C2411yescharacteristic
117C8×C197—21no—
118C8×C197C2×D412yescomplemented
119C22×C197—22no—
120C22×C197—22no—
121C22×C197—41no—
122C22×C197—41no—
123C22×C197—41no—
124C22×C197—41no—
125C22×C197C22×D411yescharacteristic
126C2×D197—22no—
127C2×D197—22no—
128C2×D197—22no—
129C2×D197—42no—
130C2×D197—42no—
131C2×D197—42no—
132C2×D197—88no—
133C4×C197—21no—
134C4×C197—41no—
135C4×C197—41no—
136C4×C197AGL(1,ℤ/8)12yescomplemented
137C4×C197C22×D411yescharacteristic
138C4×C197H(5,𝔽2)11yescharacteristic
139Dic197—22no—
140Dic197—42no—
141C2×C197—42no—
142C2×C197—42no—
143C2×C197D4211yescharacteristic
144C2×C197C2×AGL(1,ℤ/8)11yescharacteristic
145C2×C197#57711yescharacteristic
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#461—1971no—
156#459—1971no—
157#450—1972no—
158#449—1972no—
159C4×D8D19711yescharacteristic, complemented
160#436—1971no—
161C2×#41—1972no—
162C22×D4—3941no—
163C22×D4—1972no—
164AGL(1,ℤ/8)—3948no—
165C2×SD16—1972no—
166C2×SD16—1972no—
167C2×SD16—1972no—
168C2×D8—1972no—
169C2×D8C2×D19711yescharacteristic
170C2×M16—1972no—
171C4⋊Q8—1971no—
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×D4C2×D19711yescharacteristic
181C4×D4C2×D19711yescharacteristic, complemented
182C2×C4⋊C4—3941no—
183#107C2×D19711yescharacteristic, complemented
184C4⋊C8—1971no—
185#103—1972no—
186#102—1972no—
187#102C2×D19711yescharacteristic
188#102C2×D19711yescharacteristic, complemented
189C22⋊C8—1972no—
190C4×C8C2×D19711yescharacteristic, complemented
191C24—3942no—
192#41—7882no—
193#41—3944no—
194C2×Q8—1972no—
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×D4C22×D19711yescharacteristic
206C2×D4C22×D19711yescharacteristic
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—3942no—
210C22×C4—1972no—
211SD16—7882no—
212SD16—3944no—
213SD16—3944no—
214D8—3944no—
215D8D78814yescomplemented
216M16—3944no—
217C2×C8—1972no—
218C2×C8C22×D19711yescharacteristic
219C2×C8C22×D19711yescharacteristic
220C4⋊C4—3942no—
221C4⋊C4—3941no—
222C4⋊C4—3941no—
223C4⋊C4—1971no—
224C4⋊C4C22×D19711yescharacteristic
225C4⋊C4C22×D19711yescharacteristic
226C22⋊C4—21no—
227C22⋊C4—21no—
228C22⋊C4—3942no—
229C22⋊C4—3942no—
230C22⋊C4—1972no—
231C42C22×D19711yescharacteristic
232C23—21no—
233C23—21no—
234C23—7884no—
235C23—7884no—
236C23—15762no—
237C23—3942no—
238C23—1972no—
239Q8—3942no—
240Q8—3942no—
241D4—21no—
242D4—21no—
243D4—7884no—
244D4—7884no—
245D4—7884no—
246D4—7882no—
247D4—15761no—
248D4—3942no—
249D4—3942no—
250D4—3942no—
251D4C2×D78812yes—
252D4C2×D78812yescomplemented
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—7884no—
258C2×C4—3942no—
259C2×C4—3942no—
260C2×C4—3942no—
261C2×C4—1972no—
262C2×C4D4×D19711yescharacteristic
263C2×C4D4×D19711yescharacteristic
264C2×C4C23×D19711yescharacteristic
265C8—21no—
266C8—3942no—
267C8C2×D78812yescomplemented
268C22—22no—
269C22—22no—
270C22—41no—
271C22—41no—
272C22—41no—
273C22—41no—
274C22—7882no—
275C22—7882no—
276C22—7882no—
277C22—15768no—
278C22—3942no—
279C22—3942no—
280C22—3942no—
281C22C2×D4×D19711yescharacteristic
282C4—21no—
283C4—41no—
284C4—41no—
285C4—7882no—
286C4—3942no—
287C4#2026335012yescomplemented
288C4C22×D78811yescharacteristic
289C4#2026337211yescharacteristic
290C2—42no—
291C2—42no—
292C2—7882no—
293C2—15762no—
294C2D4×D78811yescharacteristic
295C2C2×#2026335011yescharacteristic
296C2#2716188611yescharacteristic
subgroupG/H
centerC22C2×D4×D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
FittingC4×D8×C197C2
socleC22×C197C22×D4
hypercenterC4×D8D197
hypocenterC197#3272
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C42 ◁ C4×D8

Complements

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

NH ≅ G/Ntype
1. C2×#20263350C2semidirect
2. #27161733C2semidirect
3. D4×D788C2semidirect
4. #27161452C2semidirect
5. #27161326C2semidirect
6. #27161284C2semidirect
7. #27161274C2semidirect
8. #27161231C2semidirect
9. #27161225C2semidirect
10. #27161158C2semidirect
11. C4×D8×C197C2semidirect
12. C2×#15691357C22semidirect
16. C2×#15691347C22semidirect
17. C2×#15691347C22semidirect
19. C2×#15691343C22semidirect
20. C2×#15691341C22semidirect
21. #20263326C22semidirect
24. #20263306C22semidirect
31. C4⋊#13096802C22semidirect
32. #20263249C22semidirect
33. #20263246C22semidirect
34. #20263238C22semidirect
37. C4×D4×C197C22semidirect
38. #107×C197C22semidirect
40. #102×C197C22semidirect
41. C4×C8×C197C22semidirect
75. D8×C197D4semidirect
109. D4×C197C2×D4semidirect
118. C8×C197C2×D4semidirect
136. C4×C197AGL(1,ℤ/8)semidirect
148. C197#3272semidirect
159. C4×D8D197semidirect
181. C4×D4C2×D197semidirect
183. #107C2×D197semidirect
188. #102C2×D197semidirect
190. C4×C8C2×D197semidirect
215. D8D788semidirect
252. D4C2×D788semidirect
267. C8C2×D788semidirect
287. C4#20263350semidirect

Maximal subgroups

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

subgroupindex
C2×#202633502
#271617332
D4×D7882
#271614522
#271613262
#271612842
#271612742
#271612312
#271612252
#271611582
C4×D8×C1972
#3272197