Finite Group Databasebeta

← #38752000

Subgroups — #38752000

Order 25216.

732 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 732 conjugacy classes of subgroups — too many to draw. Only its 149 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×#20263373C211yesmaximal, characteristic, complemented
2C2×#20263354C211yesmaximal, characteristic, complemented
3C22×#15691347C211yesmaximal, characteristic, complemented
4#27161643C211yesmaximal, characteristic, complemented
5#27161226C214yesmaximal, complemented
6#27161220C214yesmaximal, complemented
7#27161089C211yesmaximal, characteristic, complemented
8#27161076C211yesmaximal, characteristic, complemented
9#408×C197C211yesmaximal, characteristic, complemented
10#20263373—24no—
11#20263373—24no—
12C2×#15691358C2211yescharacteristic
13C2×#15691358C2212yescomplemented
14C2×#15691357—22no—
15C2×#15691357—21no—
16C22×Dic394C2211yescharacteristic
17C22×Dic394C2212yescomplemented
18C2×Q8×D197—21no—
19C22×D788C2211yescharacteristic
20#20263354—24no—
21C2×#15691347—24no—
22C2×#15691347C2212yes—
23C2×#15691347C2214yes—
24C2×D1576C2212yes—
25#20263322—21no—
26#20263315—21no—
27#20263300C2214yes—
28#20263294—22no—
29#20263285C2211yescharacteristic
30#20263251C2214yescomplemented
31#20263251C2212yescomplemented
32#20263248C2212yescomplemented
33C2×#41×C197C2211yescharacteristic
34C2×M16×C197C2211yescharacteristic
35C22×C8×C197C2211yescharacteristic
36C22⋊C8×C197C2214yescomplemented
37C23×D197—21no—
38#15691358—24no—
39#15691358—24no—
40#15691358—22no—
41#15691358D414yescomplemented
42#15691357—216no—
43#15691357—28no—
44C2×Dic394—24no—
45C2×Dic394—24no—
46C2×Dic394—21no—
47C2×Dic394—22no—
48C2×Dic394—22no—
49C2×Dic394D414yescomplemented
50C2×Dic394C2312yes—
51C2×Dic394C2311yescharacteristic
52Q8×D197—28no—
53C2×D788—22no—
54C2×D788—22no—
55C2×D788—21no—
56C2×D788C2312yes—
57C2×D788C2311yescharacteristic
58C2×C22⋊D197—21no—
59C2×C22⋊D197—22no—
60C2×C22⋊D197—24no—
61C22×Dic197—21no—
62C22×Dic197—22no—
63C2×C4×D197—21no—
64C2×C4×D197—22no—
65#15691347—28no—
66#15691347—24no—
67D1576—24no—
68C4⋊Dic197C2312yes—
69C22⋊Dic197—21no—
70#15691335—24no—
71C4×Dic197—21no—
72#41×C197—22no—
73#41×C197—22no—
74C2×Q8×C197D411yescharacteristic
75C2×D4×C197D412yes—
76C2×D4×C197D411yescharacteristic
77C22×C4×C197C2312yes—
78C22×C4×C197C2311yescharacteristic
79M16×C197—22no—
80C2×C8×C197—22no—
81C2×C8×C197C2312yes—
82C2×C8×C197C2312yes—
83C22×D197—21no—
84C22×D197—22no—
85C22×D197—42no—
86C22×D197—82no—
87Dic394—28no—
88Dic394—24no—
89Dic394—21no—
90Dic394—22no—
91Dic394C2×D412yes—
92Dic394C2×D414yes—
93D788—21no—
94D788—22no—
95D788—22no—
96D788—42no—
97D788C2×D412yes—
98C22⋊D197—28no—
99C22⋊D197—24no—
100C22⋊D197—84no—
101C2×Dic197—28no—
102C2×Dic197—24no—
103C2×Dic197—21no—
104C2×Dic197—22no—
105C2×Dic197—22no—
106C4×D197—28no—
107C4×D197—24no—
108C23×C197C2×D412yes—
109C23×C197C2×D411yescharacteristic
110Q8×C197—22no—
111D4×C197—24no—
112D4×C197—22no—
113C2×C4×C197—22no—
114C2×C4×C197—22no—
115C2×C4×C197C2×D412yes—
116C2×C4×C197C2×D412yes—
117C2×C4×C197C2×D412yes—
118C2×C4×C197C2×D411yescharacteristic
119C2×C4×C197C2411yescharacteristic
120C8×C197—22no—
121C8×C197—22no—
122C22×C197—22no—
123C22×C197—22no—
124C22×C197—24no—
125C22×C197#12012yes—
126C22×C197C22×D411yescharacteristic
127C2×D197—22no—
128C2×D197—42no—
129C2×D197—42no—
130C2×D197—44no—
131C2×D197—84no—
132C4×C197—22no—
133C4×C197#12012yes—
134C4×C197C22×D411yescharacteristic
135C4×C197C22×D411yescharacteristic
136Dic197—24no—
137Dic197—22no—
138Dic197—42no—
139C2×C197—22no—
140C2×C197—42no—
141C2×C197C2×#12011yescharacteristic
142C2×C197#57712yes—
143D197—42no—
144D197—82no—
145C197#301011yescharacteristic, complemented, Sylow
146#3010—1971nomaximal, Sylow
147C2×2−1+4—1971no—
148C2×AGL(1,ℤ/8)—1971no—
149C22×SD16—1971no—
150#535—1971no—
151#449—1974no—
152#448—1974no—
153#419—1971no—
154#416—1971no—
155#408D19711yescharacteristic, complemented
1562−1+4—3944no—
1572−1+4—3944no—
158C2×#41—3942no—
159C2×#41—3941no—
160C2×#41—1972no—
161C2×#41—1971no—
162C2×#41C2×D19711yescharacteristic
163C22×Q8—3941no—
164C22×Q8—1972no—
165C22×Q8—1971no—
166C22×D4—1971no—
167AGL(1,ℤ/8)—3944no—
168C2×SD16—3944no—
169C2×SD16—1974no—
170C2×SD16—1972no—
171C2×D8—1972no—
172C2×M16C2×D19711yescharacteristic
173C22×C8C2×D19711yescharacteristic
174#127—3941no—
175#124—3941no—
176#121—1974no—
177#120—3942no—
178#117—1971no—
179#103—1972no—
180#103—1974no—
181#102—1972no—
182C22⋊C8C2×D19714yescomplemented
183C24—3941no—
184#41—22no—
185#41—22no—
186#41—3944no—
187#41—3944no—
188#41—39416no—
189#41—3948no—
190#41—3942no—
191#41—1974no—
192C2×Q8—3944no—
193C2×Q8—3944no—
194C2×Q8—3948no—
195C2×Q8—3942no—
196C2×Q8—3942no—
197C2×Q8—3941no—
198C2×Q8—1974no—
199C2×Q8—1971no—
200C2×Q8—1972no—
201C2×Q8D78811yescharacteristic
202C2×D4—3944no—
203C2×D4—3942no—
204C2×D4—3941no—
205C2×D4—3942no—
206C2×D4—3942no—
207C2×D4—3941no—
208C2×D4—1972no—
209C2×D4—1971no—
210C2×D4D78812yes—
211C2×D4D78811yescharacteristic
212C22×C4—3942no—
213C22×C4—3941no—
214C22×C4—3942no—
215C22×C4—3941no—
216C22×C4C22×D19712yes—
217C22×C4C22×D19711yescharacteristic
218SD16—3948no—
219SD16—3944no—
220D8—3944no—
221M16—22no—
222C2×C8—22no—
223C2×C8C22×D19712yes—
224C2×C8C22×D19712yes—
225C4⋊C4—1972no—
226C22⋊C4—3941no—
227C22⋊C4—3944no—
228C42—3941no—
229C23—7882no—
230C23—15762no—
231C23—3942no—
232C23—3941no—
233C23C2×D78812yes—
234C23C2×D78811yescharacteristic
235Q8—22no—
236Q8—3948no—
237Q8—3944no—
238Q8—3941no—
239Q8—3942no—
240Q8—1972no—
241Q8—1974no—
242D4—24no—
243D4—22no—
244D4—7882no—
245D4—15764no—
246D4—3948no—
247D4—3944no—
248D4—3942no—
249D4—3942no—
250D4—3941no—
251D4—1972no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—3948no—
255C2×C4—3942no—
256C2×C4—3948no—
257C2×C4—3944no—
258C2×C4—3942no—
259C2×C4—3941no—
260C2×C4—3944no—
261C2×C4D4×D19712yes—
262C2×C4D4×D19712yes—
263C2×C4C2×D78812yes—
264C2×C4C2×D78811yescharacteristic
265C2×C4C23×D19711yescharacteristic
266C8—22no—
267C8—22no—
268C22—22no—
269C22—24no—
270C22—22no—
271C22—7882no—
272C22—7884no—
273C22—7882no—
274C22—15764no—
275C22—3942no—
276C22#2026329612yes—
277C22C22×D78811yescharacteristic
278C4—22no—
279C4—7882no—
280C4—3944no—
281C4—3942no—
282C4#2026329612yes—
283C4C2×D4×D19711yescharacteristic
284C4C2×D4×D19711yescharacteristic
285C2—22no—
286C2—42no—
287C2—7882no—
288C2—15762no—
289C2C2×#2026329611yescharacteristic
290C2#2716189112yes—
subgroupG/H
centerC22C22×D788
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#408×C197C2
socleC22×C197C22×D4
hypercenter#408D197
hypocenterC197#3010
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C2×#41 ◁ #408

Complements

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

NH ≅ G/Ntype
1. C2×#20263373C2semidirect
2. C2×#20263354C2semidirect
3. C22×#15691347C2semidirect
4. #27161643C2semidirect
5. #27161226C2semidirect
6. #27161220C2semidirect
7. #27161089C2semidirect
8. #27161076C2semidirect
9. #408×C197C2semidirect
13. C2×#15691358C22semidirect
17. C22×Dic394C22semidirect
30. #20263251C22semidirect
31. #20263251C22semidirect
32. #20263248C22semidirect
36. C22⋊C8×C197C22semidirect
41. #15691358D4semidirect
49. C2×Dic394D4semidirect
145. C197#3010semidirect
155. #408D197semidirect
182. C22⋊C8C2×D197semidirect

Maximal subgroups

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

subgroupindex
C2×#202633732
C2×#202633542
C22×#156913472
#271616432
#271612262
#271612202
#271610892
#271610762
#408×C1972
#3010197