Finite Group Databasebeta

← #38754311

Subgroups — #38754311

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
3#27161714C211yesmaximal, characteristic, complemented
4#27161322C212yesmaximal, complemented
5#27161281C211yesmaximal, characteristic, complemented
6#27161271C211yesmaximal, characteristic, complemented
7#27161228C212yesmaximal, complemented
8#27161223C212yesmaximal, complemented
9C4×D1576C211yesmaximal, characteristic, complemented
10#27161158C211yesmaximal, characteristic, complemented
11#492×C197C211yesmaximal, characteristic, complemented
12C2×#15691359C2212yescomplemented
13C2×D4×D197—21no—
14C2×D4×D197C2212yes—
15#20263353—28no—
16C2×#15691345C2212yes—
17C2×D1576C2211yescharacteristic
18C2×#15691343C2212yescomplemented
19C2×#15691341C2212yescomplemented
20#20263311—22no—
21#20263302C2212yescomplemented
22#20263301—21no—
23#20263297—21no—
24#20263297C2212yes—
25#20263294—22no—
26C4×D788C2211yescharacteristic
27C4×D788C2211yescharacteristic
28C4×D788C2211yescharacteristic, complemented
29C4⋊C4×D197—21no—
30#20263260C2211yescharacteristic, complemented
31C4⋊#13096802C2211yescharacteristic, complemented
32#20263250C2212yescomplemented
33#20263248C2211yescharacteristic
34#20263248C2211yescharacteristic, complemented
35#20263247C2212yescomplemented
36#20263238C2212yescomplemented
37C2×SD16×C197C2212yescomplemented
38C2×SD16×C197C2212yescomplemented
39C4⋊Q8×C197C2211yescharacteristic, complemented
40#127×C197C2211yescharacteristic
41C4×C8×C197C2211yescharacteristic, complemented
42C23×D197—22no—
43#15691359—24no—
44#15691359—42no—
45C2×D788—22no—
46C2×D788D411yescharacteristic
47C2×D788C2311yescharacteristic
48C2×D788C2311yescharacteristic
49C2×C22⋊D197—22no—
50C2×C22⋊D197—22no—
51D4×D197—24no—
52D4×D197—42no—
53D4×D197—44no—
54C2×C4×D197—22no—
55C2×C4×D197—21no—
56C2×C4×D197—21no—
57C2×C4×D197C2312yes—
58#15691345—24no—
59D1576D414yescomplemented
60#15691343—24no—
61#15691341—24no—
62C2×#13096802C2312yes—
63C4⋊Dic197D411yescharacteristic
64C4⋊Dic197C2311yescharacteristic
65C4⋊Dic197C2311yescharacteristic
66#15691337—22no—
67C22⋊Dic197—22no—
68#15691335—22no—
69#15691335—21no—
70#15691335—21no—
71#15691335D412yes—
72C2×Q8×C197C2312yes—
73C2×D4×C197—21no—
74C2×D4×C197—21no—
75C2×D4×C197C2312yes—
76SD16×C197—24no—
77SD16×C197—42no—
78C2×C8×C197C2311yescharacteristic
79C2×C8×C197C2311yescharacteristic
80C4⋊C4×C197—21no—
81C4⋊C4×C197—21no—
82C42×C197C2311yescharacteristic
83C22×D197—21no—
84C22×D197—21no—
85C22×D197—44no—
86C22×D197—44no—
87C22×D197—82no—
88C22×D197C2×D412yes—
89D788—22no—
90D788—21no—
91D788—21no—
92D788—44no—
93D788C2×D412yes—
94D788C2×D412yescomplemented
95C22⋊D197—44no—
96C22⋊D197—44no—
97C2×Dic197—21no—
98C2×Dic197—21no—
99C2×Dic197C2×D412yes—
100C4×D197—22no—
101C4×D197—22no—
102C4×D197—44no—
103C4×D197—42no—
104C4×D197—42no—
105#13096802—22no—
106C23×C197—22no—
107Q8×C197—22no—
108Q8×C197—22no—
109D4×C197—22no—
110D4×C197—22no—
111D4×C197—42no—
112D4×C197—81no—
113C2×C4×C197—22no—
114C2×C4×C197C2×D411yescharacteristic
115C2×C4×C197C2×D411yescharacteristic
116C2×C4×C197C2411yescharacteristic
117C8×C197—21no—
118C8×C197C2×D412yescomplemented
119C22×C197—42no—
120C22×C197—42no—
121C22×C197—42no—
122C22×C197C22×D411yescharacteristic
123C2×D197—22no—
124C2×D197—22no—
125C2×D197—22no—
126C2×D197—22no—
127C2×D197—22no—
128C2×D197—41no—
129C2×D197—41no—
130C2×D197—41no—
131C2×D197—41no—
132C2×D197—88no—
133C4×C197—21no—
134C4×C197—42no—
135C4×C197AGL(1,ℤ/8)12yescomplemented
136C4×C197C22×D411yescharacteristic
137C4×C197H(5,𝔽2)11yescharacteristic
138Dic197—22no—
139Dic197—41no—
140Dic197—41no—
141C2×C197—82no—
142C2×C197D4211yescharacteristic
143C2×C197C2×AGL(1,ℤ/8)11yescharacteristic
144C2×C197#57711yescharacteristic
145D197—42no—
146D197—42no—
147D197—42no—
148C197#327211yescharacteristic, complemented, Sylow
149#3272—1971nomaximal, Sylow
150C2×AGL(1,ℤ/8)—1972no—
151#547—1971no—
152D42—1971no—
153#492D19711yescharacteristic, complemented
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—3941no—
163C22×D4—1972no—
164AGL(1,ℤ/8)—3948no—
165C2×SD16—1972no—
166C2×SD16C2×D19712yescomplemented
167C2×SD16C2×D19712yescomplemented
168C2×D8—1972no—
169C2×D8—1971no—
170C2×M16—1972no—
171C4⋊Q8C2×D19711yescharacteristic, complemented
172#127C2×D19711yescharacteristic
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×D4—1971no—
182C2×C4⋊C4—3941no—
183#107—1971no—
184C4⋊C8—1971no—
185#103—1972no—
186#102—1972no—
187#102—1971no—
188#102—1971no—
189C22⋊C8—1972no—
190C4×C8C2×D19711yescharacteristic, complemented
191C24—3942no—
192#41—7882no—
193#41—3944no—
194C2×Q8C22×D19712yes—
195C2×D4—21no—
196C2×D4—21no—
197C2×D4—7884no—
198C2×D4—7882no—
199C2×D4—3942no—
200C2×D4—3944no—
201C2×D4—3942no—
202C2×D4—3942no—
203C2×D4—1971no—
204C2×D4—1971no—
205C2×D4—1971no—
206C2×D4C22×D19712yes—
207C22×C4—3942no—
208C22×C4—3941no—
209C22×C4—3941no—
210C22×C4—1972no—
211SD16—24no—
212SD16—42no—
213SD16—3944no—
214D8—3944no—
215D8—1974no—
216M16—3944no—
217C2×C8—1972no—
218C2×C8C22×D19711yescharacteristic
219C2×C8C22×D19711yescharacteristic
220C4⋊C4—21no—
221C4⋊C4—21no—
222C4⋊C4—3942no—
223C4⋊C4—1971no—
224C4⋊C4—1971no—
225C4⋊C4—1971no—
226C22⋊C4—3942no—
227C22⋊C4—3942no—
228C22⋊C4—3941no—
229C22⋊C4—3941no—
230C22⋊C4—1972no—
231C42C22×D19711yescharacteristic
232C23—22no—
233C23—7884no—
234C23—7884no—
235C23—15762no—
236C23—3941no—
237C23—3941no—
238C23—1972no—
239Q8—22no—
240Q8—22no—
241D4—22no—
242D4—22no—
243D4—42no—
244D4—81no—
245D4—7884no—
246D4—7884no—
247D4—7884no—
248D4—3942no—
249D4—3941no—
250D4—3941no—
251D4—1972no—
252D4—1972no—
253C2×C4—22no—
254C2×C4—7884no—
255C2×C4—7882no—
256C2×C4—7882no—
257C2×C4—3941no—
258C2×C4—3941no—
259C2×C4—3942no—
260C2×C4—3942no—
261C2×C4—1972no—
262C2×C4D4×D19711yescharacteristic
263C2×C4D4×D19711yescharacteristic
264C2×C4C23×D19711yescharacteristic
265C8—21no—
266C8—3942no—
267C8D4×D19712yescomplemented
268C22—42no—
269C22—42no—
270C22—42no—
271C22—7881no—
272C22—7881no—
273C22—7881no—
274C22—7881no—
275C22—15768no—
276C22—3942no—
277C22—3942no—
278C22—3942no—
279C22—3942no—
280C22—3942no—
281C22C2×D4×D19711yescharacteristic
282C4—21no—
283C4—42no—
284C4—7881no—
285C4—7881no—
286C4—3942no—
287C4#2026335312yescomplemented
288C4C2×D4×D19711yescharacteristic
289C4#2026337111yescharacteristic
290C2—82no—
291C2—7882no—
292C2—7882no—
293C2—7882no—
294C2#2716171411yescharacteristic
295C2C2×#2026335311yescharacteristic
296C2#2716188511yescharacteristic
subgroupG/H
centerC22C2×D4×D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#492×C197C2
socleC22×C197C22×D4
hypercenter#492D197
hypocenterC197#3272
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C42 ◁ #492

Complements

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

NH ≅ G/Ntype
1. C2×#20263353C2semidirect
2. #27161732C2semidirect
3. #27161714C2semidirect
4. #27161322C2semidirect
5. #27161281C2semidirect
6. #27161271C2semidirect
7. #27161228C2semidirect
8. #27161223C2semidirect
9. C4×D1576C2semidirect
10. #27161158C2semidirect
11. #492×C197C2semidirect
12. C2×#15691359C22semidirect
18. C2×#15691343C22semidirect
19. C2×#15691341C22semidirect
21. #20263302C22semidirect
28. C4×D788C22semidirect
30. #20263260C22semidirect
31. C4⋊#13096802C22semidirect
32. #20263250C22semidirect
34. #20263248C22semidirect
35. #20263247C22semidirect
36. #20263238C22semidirect
37. C2×SD16×C197C22semidirect
38. C2×SD16×C197C22semidirect
39. C4⋊Q8×C197C22semidirect
41. C4×C8×C197C22semidirect
59. D1576D4semidirect
94. D788C2×D4semidirect
118. C8×C197C2×D4semidirect
135. C4×C197AGL(1,ℤ/8)semidirect
148. C197#3272semidirect
153. #492D197semidirect
166. C2×SD16C2×D197semidirect
167. C2×SD16C2×D197semidirect
171. C4⋊Q8C2×D197semidirect
190. C4×C8C2×D197semidirect
267. C8D4×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
#271617142
#271613222
#271612812
#271612712
#271612282
#271612232
C4×D15762
#271611582
#492×C1972
#3272197