Finite Group Databasebeta

← #27152184

Subgroups — #27152184

Order 12416.

656 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 656 conjugacy classes of subgroups — too many to draw. Only its 153 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×#15690836C211yesmaximal, characteristic, complemented
3C2×#15690835C211yesmaximal, characteristic, complemented
4#20261281C211yesmaximal, characteristic, complemented
5C2×#15690806C211yesmaximal, characteristic, complemented
6#20261123C212yesmaximal, complemented
7#20261121C212yesmaximal, complemented
8#20261110C212yesmaximal, complemented
9#20261106C212yesmaximal, complemented
10M16×Dic97C211yesmaximal, characteristic, complemented
11C2×AGL(1,ℤ/8)×C97C211yesmaximal, characteristic, complemented
12C2×#13095511C2211yescharacteristic
13C22×D388C2211yescharacteristic
14C22×C22⋊D97—22no—
15#15690838—24no—
16#15690836—24no—
17#15690835—24no—
18C2×#13095500C2212yescomplemented
19C2×#13095499C2212yescomplemented
20C2×#13095498C2212yescomplemented
21C2×D776C2212yescomplemented
22C2×#13095496C2212yescomplemented
23C2×#13095496C2212yescomplemented
24C2×#13095495C2211yescharacteristic
25#15690808C2211yescharacteristic, complemented
26#15690806—22no—
27#15690806C2212yes—
28#15690806C2212yes—
29#15690806C2211yescharacteristic, complemented
30#15690799C2211yescharacteristic, complemented
31#15690797C2211yescharacteristic, complemented
32#15690782—22no—
33C4×C22⋊D97—22no—
34C2×C4×Dic97C2211yescharacteristic
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×D388C2311yescharacteristic
51C2×D388C2312yes—
52C2×C22⋊D97—21no—
53C2×C22⋊D97—22no—
54C2×C22⋊D97—24no—
55C2×C22⋊D97—24no—
56C2×C22⋊D97—48no—
57C22×Dic97D412yes—
58C2×C4×D97—21no—
59#13095500—24no—
60#13095499—24no—
61#13095498—24no—
62D776—24no—
63#13095496—24no—
64#13095496—24no—
65#13095495D414yescomplemented
66C2×#11715895C2312yes—
67#13095490—22no—
68C22⋊Dic97—22no—
69#13095488—22no—
70C4×Dic97D412yes—
71C4×Dic97C2311yescharacteristic
72C4×Dic97C2311yescharacteristic
73C24×C97—21no—
74#41×C97—22no—
75#41×C97—22no—
76C2×Q8×C97C2311yescharacteristic
77C2×D4×C97—21no—
78C2×D4×C97—22no—
79C2×D4×C97—22no—
80C2×D4×C97—21no—
81C2×D4×C97C2311yescharacteristic
82C2×D4×C97C2312yes—
83C22×C4×C97—21no—
84C22×C4×C97C2311yescharacteristic
85SD16×C97—24no—
86D8×C97—24no—
87M16×C97D414yescomplemented
88C2×C8×C97C2312yes—
89C23×C97—21no—
90C23×C97—22no—
91C23×C97—44no—
92C23×C97—42no—
93C23×C97C2×D411yescharacteristic
94C22×D97—21no—
95C22×D97—22no—
96C22×D97—44no—
97C22×D97—42no—
98Q8×C97—21no—
99Q8×C97—21no—
100D4×C97—21no—
101D4×C97—22no—
102D4×C97—22no—
103D4×C97—21no—
104D4×C97—42no—
105D4×C97—42no—
106C2×C4×C97—21no—
107C2×C4×C97—42no—
108C2×C4×C97C2×D411yescharacteristic
109C2×C4×C97C2×D412yes—
110C2×C4×C97C2×D412yes—
111C2×C4×C97C2411yescharacteristic
112C22⋊D97—48no—
113C22⋊D97—42no—
114C22⋊D97—44no—
115C22⋊D97—48no—
116C2×Dic97—24no—
117C2×Dic97—21no—
118C2×Dic97C2×D412yes—
119C2×Dic97C2×D412yes—
120D388—21no—
121D388—22no—
122D388—21no—
123D388—22no—
124D388—42no—
125C4×D97—42no—
126Dic194—21no—
127Dic194—21no—
128C8×C97C2×D414yescomplemented
129#11715895C2×D414yescomplemented
130C22×C97—22no—
131C22×C97—41no—
132C22×C97—42no—
133C22×C97—44no—
134C22×C97—42no—
135C22×C97—84no—
136C22×C97#12712yes—
137C22×C97C22×D411yescharacteristic
138C2×D97—41no—
139C2×D97—42no—
140C2×D97—44no—
141C2×D97—42no—
142C2×D97—84no—
143C4×C97—41no—
144C4×C97#12712yes—
145C4×C97C22×D411yescharacteristic
146C4×C97C22×D411yescharacteristic
147Dic97—22no—
148Dic97—41no—
149Dic97AGL(1,ℤ/8)14yescomplemented
150C2×C97—22no—
151C2×C97—81no—
152C2×C97—82no—
153C2×C97C2×#12711yescharacteristic
154C2×C97C2×AGL(1,ℤ/8)12yes—
155D97—81no—
156D97—82no—
157#3151—971nomaximal, Sylow
158C97#315111yescharacteristic, complemented, Sylow
159C2×AGL(1,ℤ/8)—971no—
160C2×AGL(1,ℤ/8)—971no—
161C2×AGL(1,ℤ/8)—971no—
162C2×AGL(1,ℤ/8)D9711yescharacteristic, complemented
163#532—971no—
164C2×#127—971no—
165#496—972no—
166#496—972no—
167#493—972no—
168#492—972no—
169C4×M16—971no—
170C2×#41—971no—
171C2×#41C2×D9711yescharacteristic
172C22×D4—971no—
173C22×D4—1942no—
174C22×D4C2×D9711yescharacteristic
175AGL(1,ℤ/8)—24no—
176AGL(1,ℤ/8)—1944no—
177AGL(1,ℤ/8)—1944no—
178AGL(1,ℤ/8)—1944no—
179C2×SD16—972no—
180C2×SD16—972no—
181C2×SD16—972no—
182C2×SD16C2×D9712yescomplemented
183C2×D8—972no—
184C2×D8—972no—
185C2×D8—972no—
186C2×D8C2×D9712yescomplemented
187C2×M16—971no—
188C2×M16C2×D9711yescharacteristic
189C4⋊Q8—971no—
190#127—972no—
191#127—972no—
192#127—971no—
193#127—1942no—
194#124—971no—
195#124—971no—
196#121—1942no—
197C4×D4—1942no—
198C2×C42—971no—
199#97—972no—
200C4×C8—972no—
201C24—21no—
202C24—1941no—
203#41—22no—
204#41—22no—
205#41—1942no—
206#41—1942no—
207C2×Q8—971no—
208C2×Q8C22×D9711yescharacteristic
209C2×D4—21no—
210C2×D4—22no—
211C2×D4—21no—
212C2×D4—22no—
213C2×D4—971no—
214C2×D4—972no—
215C2×D4—3888no—
216C2×D4—1942no—
217C2×D4—1944no—
218C2×D4—1944no—
219C2×D4—1941no—
220C2×D4—1941no—
221C2×D4—1942no—
222C2×D4—1942no—
223C2×D4C22×D9711yescharacteristic
224C2×D4C22×D9712yes—
225C22×C4—21no—
226C22×C4—972no—
227C22×C4—1941no—
228C22×C4C22×D9711yescharacteristic
229SD16—24no—
230SD16—1944no—
231SD16—1944no—
232SD16—1944no—
233D8—24no—
234D8—1944no—
235D8—1944no—
236D8—1944no—
237M16—974no—
238M16C22⋊D9714yescomplemented
239C2×C8—972no—
240C2×C8C22×D9712yes—
241C4⋊C4—1942no—
242C22⋊C4—1942no—
243C22⋊C4—1942no—
244C42—972no—
245C42—971no—
246C42—971no—
247C23—21no—
248C23—22no—
249C23—44no—
250C23—42no—
251C23—3884no—
252C23—3882no—
253C23—1941no—
254C23—1942no—
255C23D4×D9711yescharacteristic
256Q8—21no—
257Q8—21no—
258Q8—1941no—
259Q8—1941no—
260D4—21no—
261D4—22no—
262D4—21no—
263D4—22no—
264D4—42no—
265D4—42no—
266D4—3888no—
267D4—3884no—
268D4—3882no—
269D4—3888no—
270D4—3882no—
271D4—1941no—
272D4—1942no—
273D4—1941no—
274D4—1942no—
275C2×C4—21no—
276C2×C4—42no—
277C2×C4—972no—
278C2×C4—972no—
279C2×C4—3882no—
280C2×C4—1944no—
281C2×C4—1941no—
282C2×C4D4×D9711yescharacteristic
283C2×C4D4×D9712yes—
284C2×C4C2×C22⋊D9712yes—
285C2×C4C23×D9711yescharacteristic
286C8—974no—
287C8C2×C22⋊D9714yescomplemented
288C22—22no—
289C22—44no—
290C22—42no—
291C22—41no—
292C22—42no—
293C22—84no—
294C22—7764no—
295C22—3884no—
296C22—3882no—
297C22—3881no—
298C22—3882no—
299C22#1569080612yes—
300C22C2×D4×D9711yescharacteristic
301C4—41no—
302C4—974no—
303C4—3881no—
304C4—1942no—
305C4#1569080612yes—
306C4C2×D4×D9711yescharacteristic
307C4C22×C22⋊D9711yescharacteristic
308C2—22no—
309C2—82no—
310C2—81no—
311C2—7762no—
312C2—7761no—
313C2C2×#1569080611yescharacteristic
314C2AGL(1,ℤ/8)×D9712yes—
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#3151
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×#15690836C2semidirect
3. C2×#15690835C2semidirect
4. #20261281C2semidirect
5. C2×#15690806C2semidirect
6. #20261123C2semidirect
7. #20261121C2semidirect
8. #20261110C2semidirect
9. #20261106C2semidirect
10. M16×Dic97C2semidirect
11. C2×AGL(1,ℤ/8)×C97C2semidirect
18. C2×#13095500C22semidirect
19. C2×#13095499C22semidirect
20. C2×#13095498C22semidirect
21. C2×D776C22semidirect
22. C2×#13095496C22semidirect
23. C2×#13095496C22semidirect
25. #15690808C22semidirect
29. #15690806C22semidirect
30. #15690799C22semidirect
31. #15690797C22semidirect
35. #15690720C22semidirect
36. C8×Dic97C22semidirect
40. C2×SD16×C97C22semidirect
41. C2×D8×C97C22semidirect
65. #13095495D4semidirect
87. M16×C97D4semidirect
128. C8×C97C2×D4semidirect
129. #11715895C2×D4semidirect
149. Dic97AGL(1,ℤ/8)semidirect
158. C97#3151semidirect
162. C2×AGL(1,ℤ/8)D97semidirect
182. C2×SD16C2×D97semidirect
186. C2×D8C2×D97semidirect
238. M16C22⋊D97semidirect
287. C8C2×C22⋊D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156908382
C2×#156908362
C2×#156908352
#202612812
C2×#156908062
#202611232
#202611212
#202611102
#202611062
M16×Dic972
C2×AGL(1,ℤ/8)×C972
#315197