Finite Group Databasebeta

← #38753197

Subgroups — #38753197

Order 25216.

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

normal subgroups only

This group has 596 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×#20263354C211yesmaximal, characteristic, complemented
2C2×#20263346C211yesmaximal, characteristic, complemented
3C22×#15691343C211yesmaximal, characteristic, complemented
4#27161646C211yesmaximal, characteristic, complemented
5#27161643C211yesmaximal, characteristic, complemented
6#27161467C212yesmaximal, complemented
7#27161465C212yesmaximal, complemented
8#27161460C212yesmaximal, complemented
9#27161454C212yesmaximal, complemented
10#27161037C211yesmaximal, characteristic, complemented
11C2×AGL(1,ℤ/8)×C197C211yesmaximal, characteristic, complemented
12C2×#15691358C2211yescharacteristic
13C22×D788C2211yescharacteristic
14#20263354—24no—
15#20263346—24no—
16C2×#15691348C2211yescharacteristic, complemented
17C2×#15691347C2212yescomplemented
18C2×#15691346C2211yescharacteristic, complemented
19C2×#15691345C2211yescharacteristic, complemented
20C2×D1576C2212yescomplemented
21C2×#15691343—24no—
22C2×#15691343C2212yes—
23C2×#15691343C2212yes—
24C2×#15691343C2211yescharacteristic, complemented
25C22×#13096802C2211yescharacteristic
26#20263322C2212yescomplemented
27#20263322C2212yescomplemented
28#20263315C2212yescomplemented
29#20263313C2212yescomplemented
30#20263300—22no—
31#20263297—22no—
32#20263295—22no—
33#20263294—22no—
34#20263285C2211yescharacteristic
35#20263236C2212yescomplemented
36C8×Dic197C2212yescomplemented
37C2×#41×C197C2211yescharacteristic
38C22×D4×C197C2211yescharacteristic
39AGL(1,ℤ/8)×C197—24no—
40C2×SD16×C197C2212yescomplemented
41C2×D8×C197C2212yescomplemented
42C2×M16×C197C2211yescharacteristic
43C23×D197—21no—
44#15691358—22no—
45#15691358—22no—
46C2×Dic394C2311yescharacteristic
47C2×D788—22no—
48C2×D788—22no—
49C2×D788—21no—
50C2×D788C2312yes—
51C2×D788C2311yescharacteristic
52C2×C22⋊D197—24no—
53C2×C22⋊D197—24no—
54C2×C22⋊D197—21no—
55C2×C4×D197—21no—
56#15691348—22no—
57#15691347—24no—
58#15691346—22no—
59#15691345—22no—
60D1576—24no—
61#15691343—24no—
62#15691343—24no—
63#15691343—22no—
64C2×#13096802D414yescomplemented
65C2×#13096802C2311yescharacteristic
66C2×#13096802C2311yescharacteristic
67C4⋊Dic197D412yes—
68C22⋊Dic197—24no—
69C22⋊Dic197D412yes—
70#15691335—24no—
71C4×Dic197C2312yes—
72C24×C197—21no—
73#41×C197—22no—
74#41×C197—22no—
75C2×Q8×C197C2311yescharacteristic
76C2×D4×C197—21no—
77C2×D4×C197—22no—
78C2×D4×C197—22no—
79C2×D4×C197—21no—
80C2×D4×C197C2311yescharacteristic
81C2×D4×C197C2312yes—
82C22×C4×C197—21no—
83C22×C4×C197C2311yescharacteristic
84SD16×C197—24no—
85D8×C197—24no—
86M16×C197D414yescomplemented
87C2×C8×C197C2312yes—
88C22×D197—22no—
89C22×D197—21no—
90C22×D197—42no—
91C22×D197—82no—
92Dic394—21no—
93Dic394—21no—
94D788—22no—
95D788—21no—
96D788—21no—
97D788—22no—
98D788—42no—
99C22⋊D197—42no—
100C22⋊D197—84no—
101C22⋊D197—84no—
102C2×Dic197—21no—
103C2×Dic197C2×D414yes—
104C4×D197—42no—
105#13096802C2×D412yes—
106#13096802C2×D412yes—
107C23×C197—21no—
108C23×C197—22no—
109C23×C197—42no—
110C23×C197—82no—
111C23×C197C2×D411yescharacteristic
112Q8×C197—21no—
113Q8×C197—21no—
114D4×C197—21no—
115D4×C197—22no—
116D4×C197—22no—
117D4×C197—21no—
118D4×C197—42no—
119D4×C197—42no—
120C2×C4×C197—21no—
121C2×C4×C197—42no—
122C2×C4×C197C2×D411yescharacteristic
123C2×C4×C197C2×D412yes—
124C2×C4×C197C2×D412yes—
125C2×C4×C197C2411yescharacteristic
126C8×C197C2×D414yes—
127C22×C197—22no—
128C22×C197—41no—
129C22×C197—42no—
130C22×C197—44no—
131C22×C197—42no—
132C22×C197—84no—
133C22×C197#12712yes—
134C22×C197C22×D411yescharacteristic
135C2×D197—41no—
136C2×D197—42no—
137C2×D197—42no—
138C2×D197—44no—
139C2×D197—84no—
140C4×C197—41no—
141C4×C197#12712yes—
142C4×C197C22×D411yescharacteristic
143C4×C197C22×D411yescharacteristic
144Dic197—24no—
145Dic197—41no—
146C2×C197—22no—
147C2×C197—81no—
148C2×C197—82no—
149C2×C197C2×#12711yescharacteristic
150C2×C197#57612yes—
151D197—81no—
152D197—82no—
153C197#315511yescharacteristic, complemented, Sylow
154#3155—1971nomaximal, Sylow
155C2×AGL(1,ℤ/8)—1971no—
156C2×AGL(1,ℤ/8)D19711yescharacteristic, complemented
157C2×#135—1971no—
158C22×D8—1971no—
159#535—1971no—
160#535—1971no—
161#496—1972no—
162#496—1972no—
163#495—1972no—
164#493—1972no—
165#405—1971no—
166C2×#41—1971no—
167C2×#41C2×D19711yescharacteristic
168C22×D4—1971no—
169C22×D4C2×D19711yescharacteristic
170AGL(1,ℤ/8)—24no—
171AGL(1,ℤ/8)—3944no—
172#135—3944no—
173C2×Q16—1971no—
174C2×SD16—1971no—
175C2×SD16—1971no—
176C2×SD16—1972no—
177C2×SD16C2×D19712yescomplemented
178C2×D8—3944no—
179C2×D8—1972no—
180C2×D8—1972no—
181C2×D8—1971no—
182C2×D8—1972no—
183C2×D8C2×D19712yescomplemented
184C2×M16C2×D19711yescharacteristic
185C22×C8—1971no—
186#127—1972no—
187#127—1972no—
188#124—1972no—
189#124—1972no—
190#121—3942no—
191#121—3942no—
192#120—3942no—
193#120—3942no—
194#117—1971no—
195#97—1972no—
196C4×C8—1972no—
197C24—21no—
198C24—3941no—
199#41—22no—
200#41—22no—
201#41—3942no—
202#41—3942no—
203C2×Q8—1971no—
204C2×Q8C22×D19711yescharacteristic
205C2×D4—21no—
206C2×D4—22no—
207C2×D4—21no—
208C2×D4—22no—
209C2×D4—3941no—
210C2×D4—3944no—
211C2×D4—3944no—
212C2×D4—3941no—
213C2×D4—3942no—
214C2×D4—3942no—
215C2×D4—1972no—
216C2×D4—1971no—
217C2×D4C22×D19712yes—
218C2×D4C22×D19711yescharacteristic
219C22×C4—21no—
220C22×C4—3941no—
221C22×C4C22×D19711yescharacteristic
222Q16—3942no—
223SD16—24no—
224SD16—3942no—
225SD16—3942no—
226SD16—3944no—
227D8—24no—
228D8—3944no—
229D8—3944no—
230D8—3942no—
231D8—3944no—
232M16C22⋊D19714yescomplemented
233C2×C8—1974no—
234C2×C8—1971no—
235C2×C8—1971no—
236C2×C8C22×D19712yes—
237C4⋊C4—1972no—
238C22⋊C4—3944no—
239C22⋊C4—3944no—
240C22⋊C4—1972no—
241C42—1972no—
242C23—22no—
243C23—21no—
244C23—42no—
245C23—82no—
246C23—7882no—
247C23—15762no—
248C23—3942no—
249C23—3941no—
250C23D4×D19711yescharacteristic
251Q8—21no—
252Q8—21no—
253Q8—3941no—
254Q8—3941no—
255D4—22no—
256D4—21no—
257D4—21no—
258D4—22no—
259D4—42no—
260D4—42no—
261D4—7882no—
262D4—7882no—
263D4—15764no—
264D4—15764no—
265D4—3942no—
266D4—3941no—
267D4—3941no—
268D4—3942no—
269C2×C4—21no—
270C2×C4—42no—
271C2×C4—7882no—
272C2×C4—3941no—
273C2×C4—1974no—
274C2×C4D4×D19711yescharacteristic
275C2×C4D4×D19712yes—
276C2×C4C2×C22⋊D19712yes—
277C2×C4C23×D19711yescharacteristic
278C8—1972no—
279C8—1972no—
280C8C2×C22⋊D19714yes—
281C22—22no—
282C22—41no—
283C22—42no—
284C22—42no—
285C22—44no—
286C22—84no—
287C22—7881no—
288C22—7882no—
289C22—7882no—
290C22—7884no—
291C22—15764no—
292C22#2026332212yes—
293C22C2×D4×D19711yescharacteristic
294C4—41no—
295C4—7881no—
296C4—3944no—
297C4#2026332212yes—
298C4C2×D4×D19711yescharacteristic
299C4C22×C22⋊D19711yescharacteristic
300C2—22no—
301C2—82no—
302C2—81no—
303C2—15762no—
304C2—15761no—
305C2C2×#2026332211yescharacteristic
306C2#2716188312yes—
subgroupG/H
centerC22C2×D4×D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
FittingC2×AGL(1,ℤ/8)×C197C2
socleC22×C197C22×D4
hypercenterC2×AGL(1,ℤ/8)D197
hypocenterC197#3155
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#20263354C2semidirect
2. C2×#20263346C2semidirect
3. C22×#15691343C2semidirect
4. #27161646C2semidirect
5. #27161643C2semidirect
6. #27161467C2semidirect
7. #27161465C2semidirect
8. #27161460C2semidirect
9. #27161454C2semidirect
10. #27161037C2semidirect
11. C2×AGL(1,ℤ/8)×C197C2semidirect
16. C2×#15691348C22semidirect
17. C2×#15691347C22semidirect
18. C2×#15691346C22semidirect
19. C2×#15691345C22semidirect
20. C2×D1576C22semidirect
24. C2×#15691343C22semidirect
26. #20263322C22semidirect
27. #20263322C22semidirect
28. #20263315C22semidirect
29. #20263313C22semidirect
35. #20263236C22semidirect
36. C8×Dic197C22semidirect
40. C2×SD16×C197C22semidirect
41. C2×D8×C197C22semidirect
64. C2×#13096802D4semidirect
86. M16×C197D4semidirect
153. C197#3155semidirect
156. C2×AGL(1,ℤ/8)D197semidirect
177. C2×SD16C2×D197semidirect
183. C2×D8C2×D197semidirect
232. M16C22⋊D197semidirect

Maximal subgroups

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

subgroupindex
C2×#202633542
C2×#202633462
C22×#156913432
#271616462
#271616432
#271614672
#271614652
#271614602
#271614542
#271610372
C2×AGL(1,ℤ/8)×C1972
#3155197