Finite Group Databasebeta

← #27152221

Subgroups — #27152221

Order 12416.

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×#15690838C211yesmaximal, characteristic, complemented
2C2×#15690830C211yesmaximal, characteristic, complemented
3C22×#13095496C211yesmaximal, characteristic, complemented
4#20261302C211yesmaximal, characteristic, complemented
5#20261299C211yesmaximal, characteristic, complemented
6#20261123C212yesmaximal, complemented
7#20261121C212yesmaximal, complemented
8#20261116C212yesmaximal, complemented
9#20261110C212yesmaximal, complemented
10#20260693C211yesmaximal, characteristic, complemented
11C2×AGL(1,ℤ/8)×C97C211yesmaximal, characteristic, complemented
12C2×#13095511C2211yescharacteristic
13C22×D388C2211yescharacteristic
14#15690838—24no—
15#15690830—24no—
16C2×#13095501C2211yescharacteristic, complemented
17C2×#13095500C2212yescomplemented
18C2×#13095499C2211yescharacteristic, complemented
19C2×#13095498C2211yescharacteristic, complemented
20C2×D776C2212yescomplemented
21C2×#13095496—24no—
22C2×#13095496C2212yes—
23C2×#13095496C2212yes—
24C2×#13095496C2211yescharacteristic, complemented
25C22×#11715895C2211yescharacteristic
26#15690806C2212yescomplemented
27#15690806C2212yescomplemented
28#15690799C2212yescomplemented
29#15690797C2212yescomplemented
30#15690784—22no—
31#15690781—22no—
32#15690779—22no—
33#15690778—22no—
34#15690769C2211yescharacteristic
35#15690720C2212yescomplemented
36C8×Dic97C2212yescomplemented
37C2×#41×C97C2211yescharacteristic
38C22×D4×C97C2211yescharacteristic
39AGL(1,ℤ/8)×C97—24no—
40C2×SD16×C97C2212yescomplemented
41C2×D8×C97C2212yescomplemented
42C2×M16×C97C2211yescharacteristic
43C23×D97—21no—
44#13095511—22no—
45#13095511—22no—
46C2×Dic194C2311yescharacteristic
47C2×D388—22no—
48C2×D388—22no—
49C2×D388—21no—
50C2×D388C2312yes—
51C2×D388C2311yescharacteristic
52C2×C22⋊D97—21no—
53C2×C22⋊D97—24no—
54C2×C22⋊D97—24no—
55C2×C4×D97—21no—
56#13095501—22no—
57#13095500—24no—
58#13095499—22no—
59#13095498—22no—
60D776—24no—
61#13095496—24no—
62#13095496—24no—
63#13095496—22no—
64C2×#11715895D414yescomplemented
65C2×#11715895C2311yescharacteristic
66C2×#11715895C2311yescharacteristic
67C4⋊Dic97D412yes—
68C22⋊Dic97—24no—
69C22⋊Dic97D412yes—
70#13095488—24no—
71C4×Dic97C2312yes—
72C24×C97—21no—
73#41×C97—22no—
74#41×C97—22no—
75C2×Q8×C97C2311yescharacteristic
76C2×D4×C97—21no—
77C2×D4×C97—22no—
78C2×D4×C97—22no—
79C2×D4×C97—21no—
80C2×D4×C97C2312yes—
81C2×D4×C97C2311yescharacteristic
82C22×C4×C97—21no—
83C22×C4×C97C2311yescharacteristic
84SD16×C97—24no—
85D8×C97—24no—
86M16×C97D414yescomplemented
87C2×C8×C97C2312yes—
88C23×C97—21no—
89C23×C97—22no—
90C23×C97—42no—
91C23×C97—82no—
92C23×C97C2×D411yescharacteristic
93C22×D97—22no—
94C22×D97—21no—
95C22×D97—42no—
96C22×D97—82no—
97Q8×C97—21no—
98Q8×C97—21no—
99D4×C97—21no—
100D4×C97—22no—
101D4×C97—22no—
102D4×C97—21no—
103D4×C97—42no—
104D4×C97—42no—
105C2×C4×C97—21no—
106C2×C4×C97—42no—
107C2×C4×C97C2×D411yescharacteristic
108C2×C4×C97C2×D412yes—
109C2×C4×C97C2×D412yes—
110C2×C4×C97C2411yescharacteristic
111C22⋊D97—42no—
112C22⋊D97—84no—
113C22⋊D97—84no—
114C2×Dic97—21no—
115C2×Dic97C2×D414yes—
116D388—22no—
117D388—21no—
118D388—21no—
119D388—22no—
120D388—42no—
121C4×D97—42no—
122Dic194—21no—
123Dic194—21no—
124C8×C97C2×D414yes—
125#11715895C2×D412yes—
126#11715895C2×D412yes—
127C22×C97—22no—
128C22×C97—41no—
129C22×C97—42no—
130C22×C97—42no—
131C22×C97—44no—
132C22×C97—84no—
133C22×C97#12712yes—
134C22×C97C22×D411yescharacteristic
135C2×D97—41no—
136C2×D97—42no—
137C2×D97—42no—
138C2×D97—44no—
139C2×D97—84no—
140C4×C97—41no—
141C4×C97#12712yes—
142C4×C97C22×D411yescharacteristic
143C4×C97C22×D411yescharacteristic
144Dic97—24no—
145Dic97—41no—
146C2×C97—22no—
147C2×C97—81no—
148C2×C97—82no—
149C2×C97C2×#12711yescharacteristic
150C2×C97#57612yes—
151D97—81no—
152D97—82no—
153#3155—971nomaximal, Sylow
154C97#315511yescharacteristic, complemented, Sylow
155C2×AGL(1,ℤ/8)—971no—
156C2×AGL(1,ℤ/8)D9711yescharacteristic, complemented
157C2×#135—971no—
158C22×D8—971no—
159#535—971no—
160#535—971no—
161#496—972no—
162#496—972no—
163#495—972no—
164#493—972no—
165#405—971no—
166C2×#41—971no—
167C2×#41C2×D9711yescharacteristic
168C22×D4—971no—
169C22×D4C2×D9711yescharacteristic
170AGL(1,ℤ/8)—24no—
171AGL(1,ℤ/8)—1944no—
172#135—1944no—
173C2×Q16—971no—
174C2×SD16—971no—
175C2×SD16—971no—
176C2×SD16—972no—
177C2×SD16C2×D9712yescomplemented
178C2×D8—972no—
179C2×D8—972no—
180C2×D8—971no—
181C2×D8—972no—
182C2×D8—1944no—
183C2×D8C2×D9712yescomplemented
184C2×M16C2×D9711yescharacteristic
185C22×C8—971no—
186#127—972no—
187#127—972no—
188#124—972no—
189#124—972no—
190#121—1942no—
191#121—1942no—
192#120—1942no—
193#120—1942no—
194#117—971no—
195#97—972no—
196C4×C8—972no—
197C24—21no—
198C24—1941no—
199#41—22no—
200#41—22no—
201#41—1942no—
202#41—1942no—
203C2×Q8—971no—
204C2×Q8C22×D9711yescharacteristic
205C2×D4—21no—
206C2×D4—22no—
207C2×D4—21no—
208C2×D4—22no—
209C2×D4—972no—
210C2×D4—971no—
211C2×D4—1944no—
212C2×D4—1944no—
213C2×D4—1941no—
214C2×D4—1941no—
215C2×D4—1942no—
216C2×D4—1942no—
217C2×D4C22×D9712yes—
218C2×D4C22×D9711yescharacteristic
219C22×C4—21no—
220C22×C4—1941no—
221C22×C4C22×D9711yescharacteristic
222Q16—1942no—
223SD16—24no—
224SD16—1942no—
225SD16—1942no—
226SD16—1944no—
227D8—24no—
228D8—1944no—
229D8—1944no—
230D8—1942no—
231D8—1944no—
232M16C22⋊D9714yescomplemented
233C2×C8—974no—
234C2×C8—971no—
235C2×C8—971no—
236C2×C8C22×D9712yes—
237C4⋊C4—972no—
238C22⋊C4—972no—
239C22⋊C4—1944no—
240C22⋊C4—1944no—
241C42—972no—
242C23—22no—
243C23—21no—
244C23—42no—
245C23—82no—
246C23—7762no—
247C23—3882no—
248C23—1942no—
249C23—1941no—
250C23D4×D9711yescharacteristic
251Q8—21no—
252Q8—21no—
253Q8—1941no—
254Q8—1941no—
255D4—22no—
256D4—21no—
257D4—21no—
258D4—22no—
259D4—42no—
260D4—42no—
261D4—7764no—
262D4—7764no—
263D4—3882no—
264D4—3882no—
265D4—1942no—
266D4—1941no—
267D4—1941no—
268D4—1942no—
269C2×C4—21no—
270C2×C4—42no—
271C2×C4—974no—
272C2×C4—3882no—
273C2×C4—1941no—
274C2×C4D4×D9711yescharacteristic
275C2×C4D4×D9712yes—
276C2×C4C2×C22⋊D9712yes—
277C2×C4C23×D9711yescharacteristic
278C8—972no—
279C8—972no—
280C8C2×C22⋊D9714yes—
281C22—22no—
282C22—41no—
283C22—42no—
284C22—42no—
285C22—44no—
286C22—84no—
287C22—7764no—
288C22—3881no—
289C22—3882no—
290C22—3882no—
291C22—3884no—
292C22#1569080612yes—
293C22C2×D4×D9711yescharacteristic
294C4—41no—
295C4—3881no—
296C4—1944no—
297C4#1569080612yes—
298C4C2×D4×D9711yescharacteristic
299C4C22×C22⋊D9711yescharacteristic
300C2—22no—
301C2—82no—
302C2—81no—
303C2—7762no—
304C2—7761no—
305C2C2×#1569080611yescharacteristic
306C2#2026153912yes—
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
FittingC2×AGL(1,ℤ/8)×C97C2
socleC22×C97C22×D4
hypercenterC2×AGL(1,ℤ/8)D97
hypocenterC97#3155
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
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×#15690838C2semidirect
2. C2×#15690830C2semidirect
3. C22×#13095496C2semidirect
4. #20261302C2semidirect
5. #20261299C2semidirect
6. #20261123C2semidirect
7. #20261121C2semidirect
8. #20261116C2semidirect
9. #20261110C2semidirect
10. #20260693C2semidirect
11. C2×AGL(1,ℤ/8)×C97C2semidirect
16. C2×#13095501C22semidirect
17. C2×#13095500C22semidirect
18. C2×#13095499C22semidirect
19. C2×#13095498C22semidirect
20. C2×D776C22semidirect
24. C2×#13095496C22semidirect
26. #15690806C22semidirect
27. #15690806C22semidirect
28. #15690799C22semidirect
29. #15690797C22semidirect
35. #15690720C22semidirect
36. C8×Dic97C22semidirect
40. C2×SD16×C97C22semidirect
41. C2×D8×C97C22semidirect
64. C2×#11715895D4semidirect
86. M16×C97D4semidirect
154. C97#3155semidirect
156. C2×AGL(1,ℤ/8)D97semidirect
177. C2×SD16C2×D97semidirect
183. C2×D8C2×D97semidirect
232. M16C22⋊D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156908382
C2×#156908302
C22×#130954962
#202613022
#202612992
#202611232
#202611212
#202611162
#202611102
#202606932
C2×AGL(1,ℤ/8)×C972
#315597