Finite Group Databasebeta

← #38753174

Subgroups — #38753174

Order 25216.

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

normal subgroups only

This group has 572 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×#20263359C211yesmaximal, characteristic, complemented
2C2×#20263356C211yesmaximal, characteristic, complemented
3C2×#20263352C211yesmaximal, characteristic, complemented
4#27161625C211yesmaximal, characteristic, complemented
5C2×#20263313C211yesmaximal, characteristic, complemented
6#27161470C212yesmaximal, complemented
7#27161467C212yesmaximal, complemented
8#27161462C212yesmaximal, complemented
9#27161460C212yesmaximal, complemented
10M16×Dic197C211yesmaximal, characteristic, complemented
11C2×AGL(1,ℤ/8)×C197C211yesmaximal, characteristic, complemented
12C2×#15691358C2211yescharacteristic
13C22×Dic394C2211yescharacteristic
14#20263359—24no—
15#20263356—24no—
16#20263352—24no—
17C2×Dic788C2212yescomplemented
18C2×#15691348C2212yescomplemented
19C2×#15691347C2212yescomplemented
20C2×#15691346C2212yes—
21C2×#15691346C2212yescomplemented
22C2×#15691343C2212yes—
23C2×#15691342C2211yescharacteristic
24#20263324C2211yescharacteristic, complemented
25#20263322C2211yescharacteristic, complemented
26#20263315C2211yescharacteristic, complemented
27#20263313—22no—
28#20263313C2212yes—
29#20263313C2212yes—
30#20263313C2211yescharacteristic, complemented
31#20263298—22no—
32C4×C22⋊D197—22no—
33C2×C22⋊Dic197—22no—
34C2×C4×Dic197C2211yescharacteristic
35#20263236C2212yescomplemented
36C8×Dic197C2212yescomplemented
37C2×#41×C197C2211yescharacteristic
38C22×D4×C197C2211yescharacteristic
39AGL(1,ℤ/8)×C197—24no—
40C2×SD16×C197C2212yes—
41C2×D8×C197C2212yes—
42C2×M16×C197C2211yescharacteristic
43#15691358—22no—
44#15691358—22no—
45C2×Dic394—21no—
46C2×Dic394—22no—
47C2×Dic394—22no—
48C2×Dic394C2312yes—
49C2×Dic394C2311yescharacteristic
50C2×D788C2311yescharacteristic
51C2×C22⋊D197—22no—
52C2×C22⋊D197—21no—
53C22×Dic197—21no—
54C22×Dic197D412yes—
55C2×C4×D197—21no—
56Dic788—24no—
57#15691348—24no—
58#15691347—24no—
59#15691346—24no—
60#15691346—24no—
61#15691343—24no—
62#15691342D414yescomplemented
63C2×#13096802C2312yes—
64#15691337—22no—
65C22⋊Dic197—24no—
66C22⋊Dic197—24no—
67C22⋊Dic197—22no—
68#15691335—22no—
69C4×Dic197D412yes—
70C4×Dic197C2311yescharacteristic
71C4×Dic197C2311yescharacteristic
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—21no—
89Dic394—21no—
90Dic394—22no—
91Dic394—22no—
92Dic394—21no—
93Dic394—42no—
94D788—21no—
95D788—21no—
96C22⋊D197—44no—
97C22⋊D197—42no—
98C2×Dic197—22no—
99C2×Dic197—21no—
100C2×Dic197—42no—
101C2×Dic197—42no—
102C2×Dic197C2×D412yes—
103C2×Dic197C2×D412yes—
104C4×D197—42no—
105#13096802C2×D414yes—
106C23×C197—21no—
107C23×C197—22no—
108C23×C197—44no—
109C23×C197—42no—
110C23×C197C2×D411yescharacteristic
111Q8×C197—21no—
112Q8×C197—21no—
113D4×C197—21no—
114D4×C197—22no—
115D4×C197—22no—
116D4×C197—21no—
117D4×C197—42no—
118D4×C197—42no—
119C2×C4×C197—21no—
120C2×C4×C197—42no—
121C2×C4×C197C2×D411yescharacteristic
122C2×C4×C197C2×D412yes—
123C2×C4×C197C2×D412yes—
124C2×C4×C197C2411yescharacteristic
125C8×C197C2×D414yes—
126C22×C197—22no—
127C22×C197—41no—
128C22×C197—42no—
129C22×C197—44no—
130C22×C197—42no—
131C22×C197—84no—
132C22×C197#12712yes—
133C22×C197C22×D411yescharacteristic
134C2×D197—41no—
135C2×D197—42no—
136C4×C197—41no—
137C4×C197#12712yes—
138C4×C197C22×D411yescharacteristic
139C4×C197C22×D411yescharacteristic
140Dic197—22no—
141Dic197—22no—
142Dic197—41no—
143Dic197—42no—
144C2×C197—22no—
145C2×C197—81no—
146C2×C197—82no—
147C2×C197C2×#12711yescharacteristic
148C2×C197#57512yes—
149D197—81no—
150C197#315311yescharacteristic, complemented, Sylow
151#3153—1971nomaximal, Sylow
152C2×#137—1971no—
153C2×#137—1971no—
154C2×AGL(1,ℤ/8)—1971no—
155C2×AGL(1,ℤ/8)D19711yescharacteristic, complemented
156#532—1971no—
157C2×#124—1971no—
158#497—1972no—
159#496—1972no—
160#495—1972no—
161#495—1972no—
162C4×M16—1971no—
163C2×#41—1971no—
164C2×#41C2×D19711yescharacteristic
165C22×Q8—1971no—
166C22×D4C2×D19711yescharacteristic
167#137—3944no—
168#137—3944no—
169AGL(1,ℤ/8)—24no—
170AGL(1,ℤ/8)—3944no—
171C2×Q16—1972no—
172C2×Q16—1972no—
173C2×SD16—1972no—
174C2×SD16—1972no—
175C2×SD16—1972no—
176C2×SD16C2×D19712yes—
177C2×D8—1972no—
178C2×D8C2×D19712yes—
179C2×M16—1971no—
180C2×M16C2×D19711yescharacteristic
181C4⋊Q8—1971no—
182#127—1971no—
183#124—3942no—
184#124—1971no—
185#124—1972no—
186#124—1972no—
187#124—1971no—
188#121—3942no—
189C4×D4—3942no—
190C2×C22⋊C4—3942no—
191C2×C42—1971no—
192#97—1972no—
193C4×C8—1972no—
194C24—21no—
195#41—22no—
196#41—22no—
197#41—3942no—
198#41—3942no—
199C2×Q8—3942no—
200C2×Q8—3942no—
201C2×Q8—3941no—
202C2×Q8—1972no—
203C2×Q8—1971no—
204C2×Q8C22×D19711yescharacteristic
205C2×D4—21no—
206C2×D4—22no—
207C2×D4—21no—
208C2×D4—22no—
209C2×D4—3941no—
210C2×D4—3942no—
211C2×D4—1971no—
212C2×D4C22×D19712yes—
213C2×D4C22×D19711yescharacteristic
214C22×C4—21no—
215C22×C4—3941no—
216C22×C4—3941no—
217C22×C4—1972no—
218C22×C4C22×D19711yescharacteristic
219Q16—3944no—
220Q16—3944no—
221SD16—24no—
222SD16—3944no—
223SD16—3944no—
224SD16—3944no—
225D8—24no—
226D8—3944no—
227M16—1974no—
228M16C22⋊D19714yescomplemented
229C2×C8—1972no—
230C2×C8C22×D19712yes—
231C4⋊C4—3942no—
232C22⋊C4—3942no—
233C22⋊C4—3944no—
234C22⋊C4—3944no—
235C22⋊C4—3942no—
236C42—1972no—
237C42—1971no—
238C42—1971no—
239C23—22no—
240C23—21no—
241C23—44no—
242C23—42no—
243C23—3941no—
244C23D4×D19711yescharacteristic
245Q8—21no—
246Q8—21no—
247Q8—7882no—
248Q8—3941no—
249Q8—3942no—
250Q8—3942no—
251Q8—3941no—
252D4—22no—
253D4—21no—
254D4—21no—
255D4—22no—
256D4—42no—
257D4—42no—
258D4—7882no—
259D4—7884no—
260D4—3941no—
261D4—3941no—
262C2×C4—21no—
263C2×C4—42no—
264C2×C4—7882no—
265C2×C4—7882no—
266C2×C4—7882no—
267C2×C4—3942no—
268C2×C4—3941no—
269C2×C4—1972no—
270C2×C4—1972no—
271C2×C4D4×D19711yescharacteristic
272C2×C4D4×D19712yes—
273C2×C4C2×C22⋊D19712yes—
274C2×C4C23×D19711yescharacteristic
275C8—1974no—
276C8C2×C22⋊D19714yes—
277C22—22no—
278C22—41no—
279C22—42no—
280C22—42no—
281C22—44no—
282C22—84no—
283C22—7881no—
284C22—7882no—
285C22#2026332212yes—
286C22C2×D4×D19711yescharacteristic
287C4—41no—
288C4—7881no—
289C4—7882no—
290C4—3942no—
291C4—3942no—
292C4#2026332212yes—
293C4C2×D4×D19711yescharacteristic
294C4C22×C22⋊D19711yescharacteristic
295C2—22no—
296C2—82no—
297C2—81no—
298C2—15761no—
299C2C2×#2026332211yescharacteristic
300C2#2716187512yes—
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#3153
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×#20263359C2semidirect
2. C2×#20263356C2semidirect
3. C2×#20263352C2semidirect
4. #27161625C2semidirect
5. C2×#20263313C2semidirect
6. #27161470C2semidirect
7. #27161467C2semidirect
8. #27161462C2semidirect
9. #27161460C2semidirect
10. M16×Dic197C2semidirect
11. C2×AGL(1,ℤ/8)×C197C2semidirect
17. C2×Dic788C22semidirect
18. C2×#15691348C22semidirect
19. C2×#15691347C22semidirect
21. C2×#15691346C22semidirect
24. #20263324C22semidirect
25. #20263322C22semidirect
26. #20263315C22semidirect
30. #20263313C22semidirect
35. #20263236C22semidirect
36. C8×Dic197C22semidirect
62. #15691342D4semidirect
86. M16×C197D4semidirect
150. C197#3153semidirect
155. C2×AGL(1,ℤ/8)D197semidirect
228. M16C22⋊D197semidirect

Maximal subgroups

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

subgroupindex
C2×#202633592
C2×#202633562
C2×#202633522
#271616252
C2×#202633132
#271614702
#271614672
#271614622
#271614602
M16×Dic1972
C2×AGL(1,ℤ/8)×C1972
#3153197