Finite Group Databasebeta

← #38751956

Subgroups — #38751956

Order 25216.

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

normal subgroups only

This group has 758 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
1C22×#15691357C211yesmaximal, characteristic, complemented
2C2×#20263359C211yesmaximal, characteristic, complemented
3C2×#20263352C211yesmaximal, characteristic, complemented
4#27161591C211yesmaximal, characteristic, complemented
5#27161242C214yesmaximal, complemented
6#27161225C214yesmaximal, complemented
7#27161087C211yesmaximal, characteristic, complemented
8#27161042C211yesmaximal, characteristic, complemented
9#418×C197C211yesmaximal, characteristic, complemented
10C2×#15691358C2211yescharacteristic
11C2×#15691357—24no—
12C2×#15691357—22no—
13C2×#15691357—24no—
14C2×#15691357C2214yes—
15C22×Dic394C2211yescharacteristic
16C22×C22⋊D197—21no—
17C23×Dic197—21no—
18C22×C4×D197C2211yescharacteristic
19#20263359—24no—
20#20263352—24no—
21C2×Dic788C2212yescomplemented
22C2×#15691347C2212yescomplemented
23C2×#15691346C2212yescomplemented
24C2×#15691343C2212yescomplemented
25C2×#15691342C2211yescharacteristic
26#20263310—21no—
27#20263306C2212yescomplemented
28#20263301C2212yescomplemented
29#20263296—21no—
30C4×D788—22no—
31#20263249C2214yescomplemented
32#20263238C2214yescomplemented
33C22×D4×C197C2211yescharacteristic
34C2×M16×C197C2211yescharacteristic
35#117×C197C2211yescharacteristic
36#102×C197C2214yescomplemented
37C23×D197D411yescharacteristic
38#15691358—22no—
39#15691358—22no—
40#15691357—216no—
41#15691357—28no—
42#15691357—28no—
43C2×Dic394—21no—
44C2×Dic394—22no—
45C2×Dic394D414yescomplemented
46C2×Dic394C2311yescharacteristic
47C2×Dic394C2312yes—
48C2×D788—22no—
49C2×D788C2311yescharacteristic
50C2×C22⋊D197—28no—
51C2×C22⋊D197—24no—
52C2×C22⋊D197—21no—
53C22×Dic197—28no—
54C22×Dic197—24no—
55C22×Dic197—21no—
56C22×Dic197D411yescharacteristic
57C2×C4×D197—22no—
58C2×C4×D197—22no—
59C2×C4×D197—21no—
60C2×C4×D197D412yes—
61C2×C4×D197C2312yes—
62Dic788—24no—
63#15691347—24no—
64#15691346—24no—
65#15691343—24no—
66#15691342—22no—
67C2×#13096802C2312yes—
68C4⋊Dic197—22no—
69#15691335—22no—
70#15691335—22no—
71C24×C197—21no—
72C2×D4×C197—21no—
73C2×D4×C197—22no—
74C2×D4×C197D414yescomplemented
75C2×D4×C197C2312yes—
76C22×C4×C197C2311yescharacteristic
77M16×C197—22no—
78C2×C8×C197C2312yes—
79C4⋊C4×C197C2312yes—
80C22⋊C4×C197—21no—
81C42×C197—21no—
82C22×D197—22no—
83C22×D197—21no—
84C22×D197—42no—
85C22×D197C2×D412yes—
86Dic394—24no—
87Dic394—22no—
88Dic394—21no—
89Dic394—21no—
90Dic394C2×D414yes—
91D788—21no—
92D788—21no—
93D788—44no—
94C22⋊D197—216no—
95C22⋊D197—42no—
96C2×Dic197—216no—
97C2×Dic197—22no—
98C2×Dic197—24no—
99C2×Dic197—22no—
100C2×Dic197—21no—
101C2×Dic197C2×D412yes—
102C4×D197—24no—
103C4×D197—22no—
104C4×D197—22no—
105C4×D197—42no—
106#13096802—22no—
107C23×C197—24no—
108C23×C197—22no—
109C23×C197—44no—
110C23×C197C2×D411yescharacteristic
111D4×C197—24no—
112D4×C197—22no—
113D4×C197C2×D414yes—
114C2×C4×C197—22no—
115C2×C4×C197C2×D411yescharacteristic
116C2×C4×C197C2×D412yes—
117C2×C4×C197C2×D412yes—
118C2×C4×C197C2411yescharacteristic
119C8×C197—22no—
120C22×C197—22no—
121C22×C197—24no—
122C22×C197—48no—
123C22×C197—44no—
124C22×C197#12012yes—
125C22×C197C22×D411yescharacteristic
126C2×D197—22no—
127C2×D197—22no—
128C2×D197—22no—
129C2×D197—44no—
130C2×D197—41no—
131C2×D197—42no—
132C4×C197—42no—
133C4×C197#12012yes—
134C4×C197C22×D411yescharacteristic
135C4×C197C22×D411yescharacteristic
136Dic197—22no—
137Dic197—24no—
138Dic197—41no—
139C2×C197—22no—
140C2×C197—44no—
141C2×C197C2×#12011yescharacteristic
142C2×C197#57512yes—
143D197—42no—
144D197—81no—
145C197#300511yescharacteristic, complemented, Sylow
146#3005—1971nomaximal, Sylow
147C22×#41—1971no—
148C2×#137—1971no—
149C2×AGL(1,ℤ/8)—1971no—
150#525—1971no—
151#452—1974no—
152#449—1974no—
153#419—1971no—
154#418D19711yescharacteristic, complemented
155#407—1971no—
156C2×#41—3944no—
157C2×#41—3944no—
158C2×#41—3942no—
159C2×#41—1974no—
160C2×#41—1971no—
161C22×Q8—1971no—
162C22×D4—3941no—
163C22×D4C2×D19711yescharacteristic
164C23×C4—3941no—
165C23×C4—1971no—
166#137—3944no—
167AGL(1,ℤ/8)—3944no—
168C2×Q16—1972no—
169C2×SD16—1972no—
170C2×SD16—1972no—
171C2×D8—1972no—
172C2×M16—1971no—
173C2×M16C2×D19711yescharacteristic
174#123—3941no—
175C22⋊Q8—1972no—
176#121—1972no—
177#120—3941no—
178C4×D4—3942no—
179#117C2×D19711yescharacteristic
180#103—1974no—
181#102C2×D19714yescomplemented
182C22⋊C8—1974no—
183C24—21no—
184C24—1971no—
185#41—3942no—
186#41—3942no—
187#41—39416no—
188#41—3948no—
189#41—3948no—
190C2×Q8—3941no—
191C2×Q8—3942no—
192C2×Q8—1974no—
193C2×Q8—1971no—
194C2×Q8—1972no—
195C2×D4—21no—
196C2×D4—22no—
197C2×D4—3942no—
198C2×D4—3941no—
199C2×D4—3948no—
200C2×D4—3944no—
201C2×D4—1971no—
202C2×D4D78814yescomplemented
203C2×D4C22×D19712yes—
204C22×C4—3941no—
205C22×C4—3948no—
206C22×C4—3941no—
207C22×C4—3944no—
208C22×C4—3942no—
209C22×C4—3942no—
210C22×C4—1971no—
211C22×C4—1972no—
212C22×C4—1972no—
213C22×C4C22×D19711yescharacteristic
214Q16—3944no—
215SD16—3944no—
216SD16—3944no—
217D8—3944no—
218M16—22no—
219M16—3942no—
220C2×C8—1972no—
221C2×C8C22×D19712yes—
222C4⋊C4—3942no—
223C4⋊C4C22×D19712yes—
224C22⋊C4—21no—
225C22⋊C4—3942no—
226C22⋊C4—3942no—
227C42—21no—
228C23—24no—
229C23—22no—
230C23—44no—
231C23—7882no—
232C23—3941no—
233C23—3942no—
234C23—1972no—
235C23D4×D19711yescharacteristic
236Q8—3941no—
237Q8—3941no—
238Q8—3944no—
239Q8—3942no—
240Q8—1974no—
241D4—24no—
242D4—22no—
243D4—7884no—
244D4—7882no—
245D4—39416no—
246D4—3941no—
247D4—3941no—
248D4C2×D78814yes—
249C2×C4—22no—
250C2×C4—7882no—
251C2×C4—3941no—
252C2×C4—39416no—
253C2×C4—3942no—
254C2×C4—3944no—
255C2×C4—3942no—
256C2×C4—3944no—
257C2×C4—3942no—
258C2×C4—3942no—
259C2×C4—1972no—
260C2×C4D4×D19711yescharacteristic
261C2×C4D4×D19712yes—
262C2×C4C2×D78812yes—
263C2×C4C23×D19711yescharacteristic
264C8—22no—
265C8—3942no—
266C22—22no—
267C22—24no—
268C22—48no—
269C22—44no—
270C22—7884no—
271C22—7882no—
272C22—7881no—
273C22—3942no—
274C22—3942no—
275C22—3942no—
276C22#2026329612yes—
277C22C2×D4×D19711yescharacteristic
278C4—42no—
279C4—7881no—
280C4—3942no—
281C4—3944no—
282C4#2026329612yes—
283C4C2×D4×D19711yescharacteristic
284C4C22×D78811yescharacteristic
285C2—22no—
286C2—44no—
287C2—7882no—
288C2—15761no—
289C2C2×#2026329611yescharacteristic
290C2#2716187512yes—
subgroupG/H
centerC22C2×D4×D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#418×C197C2
socleC22×C197C22×D4
hypercenter#418D197
hypocenterC197#3005
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C22×C4 ◁ #418

Complements

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

NH ≅ G/Ntype
1. C22×#15691357C2semidirect
2. C2×#20263359C2semidirect
3. C2×#20263352C2semidirect
4. #27161591C2semidirect
5. #27161242C2semidirect
6. #27161225C2semidirect
7. #27161087C2semidirect
8. #27161042C2semidirect
9. #418×C197C2semidirect
21. C2×Dic788C22semidirect
22. C2×#15691347C22semidirect
23. C2×#15691346C22semidirect
24. C2×#15691343C22semidirect
27. #20263306C22semidirect
28. #20263301C22semidirect
31. #20263249C22semidirect
32. #20263238C22semidirect
36. #102×C197C22semidirect
45. C2×Dic394D4semidirect
74. C2×D4×C197D4semidirect
145. C197#3005semidirect
154. #418D197semidirect
181. #102C2×D197semidirect
202. C2×D4D788semidirect

Maximal subgroups

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

subgroupindex
C22×#156913572
C2×#202633592
C2×#202633522
#271615912
#271612422
#271612252
#271610872
#271610422
#418×C1972
#3005197