Finite Group Databasebeta

← #38754308

Subgroups — #38754308

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×#20263352C212yesmaximal, complemented
2#27161734C211yesmaximal, characteristic, complemented
3#27161451C211yesmaximal, characteristic, complemented
4#27161323C212yesmaximal, complemented
5#27161284C211yesmaximal, characteristic, complemented
6#27161271C211yesmaximal, characteristic, complemented
7#27161230C212yesmaximal, complemented
8#27161222C212yesmaximal, complemented
9C4×#15691343C211yesmaximal, characteristic, complemented
10#27161157C211yesmaximal, characteristic, complemented
11D42×C197C211yesmaximal, characteristic, complemented
12C2×#15691358C2212yescomplemented
13#20263352—28no—
14C2×#15691346C2212yes—
15C2×#15691346C2212yescomplemented
16C2×#15691346C2212yescomplemented
17C2×#15691343C2211yescharacteristic
18C2×#15691343C2212yescomplemented
19C2×#15691342C2212yescomplemented
20#20263326C2211yescharacteristic, complemented
21#20263308—22no—
22#20263305C2212yescomplemented
23#20263297—21no—
24C4×D788C2211yescharacteristic, complemented
25C4⋊C4×D197—21no—
26#20263259C2211yescharacteristic, complemented
27C4⋊#13096802C2211yescharacteristic, complemented
28#20263249C2212yescomplemented
29#20263247C2211yescharacteristic
30#20263247C2212yescomplemented
31#20263246C2211yescharacteristic, complemented
32C22⋊#13096802C2212yescomplemented
33C4×#13096802C2211yescharacteristic, complemented
34C22×D4×C197—21no—
35C22×D4×C197C2212yes—
36#127×C197C2211yescharacteristic
37#121×C197—21no—
38#121×C197C2212yes—
39#120×C197—22no—
40C4×D4×C197C2211yescharacteristic
41C4×D4×C197C2211yescharacteristic
42#15691358—24no—
43#15691358—42no—
44C2×Dic394C2312yes—
45C2×D788C2311yescharacteristic
46C2×C22⋊D197—22no—
47C2×C4×D197—21no—
48C2×C4×D197—22no—
49#15691346—24no—
50#15691346—24no—
51#15691346—42no—
52#15691343—24no—
53#15691343D414yescomplemented
54#15691342—24no—
55C2×#13096802C2311yescharacteristic
56C2×#13096802C2312yes—
57C2×#13096802C2311yescharacteristic
58C4⋊Dic197—21no—
59C4⋊Dic197—21no—
60C4⋊Dic197C2311yescharacteristic
61#15691337—22no—
62C22⋊Dic197—22no—
63#15691335—21no—
64C24×C197—22no—
65C2×D4×C197—24no—
66C2×D4×C197—22no—
67C2×D4×C197—22no—
68C2×D4×C197—21no—
69C2×D4×C197—21no—
70C2×D4×C197—42no—
71C2×D4×C197—44no—
72C2×D4×C197D411yescharacteristic
73C2×D4×C197C2312yes—
74C2×D4×C197C2311yescharacteristic
75C22×C4×C197—21no—
76C22×C4×C197C2312yes—
77C4⋊C4×C197D411yescharacteristic
78C4⋊C4×C197C2311yescharacteristic
79C22⋊C4×C197—21no—
80C22⋊C4×C197—22no—
81C22⋊C4×C197D412yes—
82C42×C197C2311yescharacteristic
83C22×D197—21no—
84Dic394—22no—
85Dic394—22no—
86D788—21no—
87D788C2×D412yescomplemented
88C22⋊D197—44no—
89C2×Dic197—21no—
90C2×Dic197—22no—
91C4×D197—44no—
92C4×D197—42no—
93#13096802—22no—
94#13096802—21no—
95#13096802C2×D412yescomplemented
96C23×C197—21no—
97C23×C197—22no—
98C23×C197—44no—
99C23×C197—44no—
100C23×C197—82no—
101C23×C197C2×D412yes—
102D4×C197—22no—
103D4×C197—22no—
104D4×C197—21no—
105D4×C197—22no—
106D4×C197—44no—
107D4×C197—44no—
108D4×C197—42no—
109D4×C197—81no—
110D4×C197C2×D412yes—
111C2×C4×C197—22no—
112C2×C4×C197—22no—
113C2×C4×C197—21no—
114C2×C4×C197—42no—
115C2×C4×C197C2×D411yescharacteristic
116C2×C4×C197C2×D412yes—
117C2×C4×C197C2×D411yescharacteristic
118C2×C4×C197C2411yescharacteristic
119C22×C197—22no—
120C22×C197—22no—
121C22×C197—22no—
122C22×C197—22no—
123C22×C197—41no—
124C22×C197—41no—
125C22×C197—42no—
126C22×C197—42no—
127C22×C197—42no—
128C22×C197—88no—
129C22×C197C22×D411yescharacteristic
130C2×D197—22no—
131C2×D197—41no—
132C2×D197—41no—
133C4×C197—22no—
134C4×C197—21no—
135C4×C197—41no—
136C4×C197AGL(1,ℤ/8)12yescomplemented
137C4×C197C22×D411yescharacteristic
138C4×C197H(5,𝔽2)11yescharacteristic
139Dic197—42no—
140Dic197—41no—
141C2×C197—42no—
142C2×C197—42no—
143C2×C197—82no—
144C2×C197D4211yescharacteristic
145C2×C197C2×AGL(1,ℤ/8)11yescharacteristic
146C2×C197#57711yescharacteristic
147D197—42no—
148C197#327211yescharacteristic, complemented, Sylow
149#3272—1971nomaximal, Sylow
150C2×AGL(1,ℤ/8)—1972no—
151#547—1971no—
152D42D19711yescharacteristic, complemented
153#492—1971no—
154#468—1972no—
155#461—1971no—
156#459—1971no—
157#450—1972no—
158#449—1972no—
159C4×D8—1971no—
160#436—1971no—
161C2×#41—1972no—
162C22×D4—21no—
163C22×D4C2×D19712yes—
164AGL(1,ℤ/8)—3948no—
165C2×SD16—1972no—
166C2×SD16—1972no—
167C2×SD16—1972no—
168C2×D8—1972no—
169C2×D8—1971no—
170C2×M16—1972no—
171C4⋊Q8—1971no—
172#127C2×D19711yescharacteristic
173#123—3942no—
174C22⋊Q8—1972no—
175#121—21no—
176#121—3941no—
177#121C2×D19712yes—
178#120—22no—
179C4×D4—1971no—
180C4×D4C2×D19711yescharacteristic
181C4×D4C2×D19711yescharacteristic
182C2×C4⋊C4—3941no—
183#107—1971no—
184C4⋊C8—1971no—
185#103—1972no—
186#102—1971no—
187#102—1972no—
188#102—1971no—
189C22⋊C8—1972no—
190C4×C8—1971no—
191C24—22no—
192#41—7882no—
193#41—3944no—
194C2×Q8—1972no—
195C2×D4—24no—
196C2×D4—22no—
197C2×D4—22no—
198C2×D4—21no—
199C2×D4—21no—
200C2×D4—42no—
201C2×D4—44no—
202C2×D4—3942no—
203C2×D4—1971no—
204C2×D4C22⋊D19711yescharacteristic
205C2×D4C22×D19712yes—
206C2×D4C22×D19711yescharacteristic
207C22×C4—21no—
208C22×C4—3941no—
209C22×C4—3942no—
210C22×C4C22×D19712yes—
211SD16—7882no—
212SD16—3944no—
213SD16—3944no—
214D8—3944no—
215D8—1974no—
216M16—3944no—
217C2×C8—1971no—
218C2×C8—1972no—
219C2×C8—1971no—
220C4⋊C4—3942no—
221C4⋊C4—3941no—
222C4⋊C4—3941no—
223C4⋊C4—1971no—
224C4⋊C4C22⋊D19711yescharacteristic
225C4⋊C4C22×D19711yescharacteristic
226C22⋊C4—21no—
227C22⋊C4—22no—
228C22⋊C4—3942no—
229C22⋊C4—3941no—
230C22⋊C4C22⋊D19712yes—
231C42C22×D19711yescharacteristic
232C23—21no—
233C23—22no—
234C23—44no—
235C23—44no—
236C23—82no—
237C23—3941no—
238C23C2×C22⋊D19712yes—
239Q8—3942no—
240Q8—3942no—
241D4—22no—
242D4—22no—
243D4—21no—
244D4—22no—
245D4—44no—
246D4—44no—
247D4—42no—
248D4—81no—
249D4—7884no—
250D4—3941no—
251D4—1972no—
252D4D4×D19712yes—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—21no—
256C2×C4—42no—
257C2×C4—7884no—
258C2×C4—7882no—
259C2×C4—3941no—
260C2×C4—3942no—
261C2×C4C2×C22⋊D19711yescharacteristic
262C2×C4C2×C22⋊D19712yes—
263C2×C4C2×C22⋊D19711yescharacteristic
264C2×C4C23×D19711yescharacteristic
265C8—3942no—
266C8—3941no—
267C8—1972no—
268C22—22no—
269C22—22no—
270C22—22no—
271C22—22no—
272C22—41no—
273C22—41no—
274C22—42no—
275C22—42no—
276C22—42no—
277C22—88no—
278C22—7881no—
279C22—7881no—
280C22—3942no—
281C22C22×C22⋊D19711yescharacteristic
282C4—21no—
283C4—22no—
284C4—41no—
285C4—7881no—
286C4—7882no—
287C4#2026335212yescomplemented
288C4C2×D4×D19711yescharacteristic
289C4#2026337111yescharacteristic
290C2—42no—
291C2—42no—
292C2—82no—
293C2—7882no—
294C2D4×C22⋊D19711yescharacteristic
295C2C2×#2026335211yescharacteristic
296C2#2716189011yescharacteristic
subgroupG/H
centerC22C22×C22⋊D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
FittingD42×C197C2
socleC22×C197C22×D4
hypercenterD42D197
hypocenterC197#3272
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C4×D4 ◁ D42

Complements

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

NH ≅ G/Ntype
1. C2×#20263352C2semidirect
2. #27161734C2semidirect
3. #27161451C2semidirect
4. #27161323C2semidirect
5. #27161284C2semidirect
6. #27161271C2semidirect
7. #27161230C2semidirect
8. #27161222C2semidirect
9. C4×#15691343C2semidirect
10. #27161157C2semidirect
11. D42×C197C2semidirect
12. C2×#15691358C22semidirect
15. C2×#15691346C22semidirect
16. C2×#15691346C22semidirect
18. C2×#15691343C22semidirect
19. C2×#15691342C22semidirect
20. #20263326C22semidirect
22. #20263305C22semidirect
24. C4×D788C22semidirect
26. #20263259C22semidirect
27. C4⋊#13096802C22semidirect
28. #20263249C22semidirect
30. #20263247C22semidirect
31. #20263246C22semidirect
32. C22⋊#13096802C22semidirect
33. C4×#13096802C22semidirect
53. #15691343D4semidirect
87. D788C2×D4semidirect
95. #13096802C2×D4semidirect
136. C4×C197AGL(1,ℤ/8)semidirect
148. C197#3272semidirect
152. D42D197semidirect
287. C4#20263352semidirect

Maximal subgroups

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

subgroupindex
C2×#202633522
#271617342
#271614512
#271613232
#271612842
#271612712
#271612302
#271612222
C4×#156913432
#271611572
D42×C1972
#3272197