Finite Group Databasebeta

← #496×D97 #27152166

Subgroups — #496×D97 #27152166

Order 12416.

668 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 668 conjugacy classes of subgroups — too many to draw. Only its 145 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×SD16×D97C212yesmaximal, complemented
2C2×D8×D97C212yesmaximal, complemented
3#127×D97C211yesmaximal, characteristic, complemented
4#124×D97C211yesmaximal, characteristic, complemented
5#20261125C211yesmaximal, characteristic, complemented
6#20261124C211yesmaximal, characteristic, complemented
7#20261123C212yesmaximal, complemented
8#20261122C211yesmaximal, characteristic, complemented
9#20261121C212yesmaximal, complemented
10#97×D97C211yesmaximal, characteristic, complemented
11#496×C97C211yesmaximal, characteristic, complemented
12C2×Q8×D97C2211yescharacteristic
13C2×D4×D97—22no—
14C2×D4×D97C2211yescharacteristic
15C2×D4×D97C2212yes—
16C2×#13095500C2212yescomplemented
17C2×#13095499C2212yescomplemented
18C2×#13095498C2212yescomplemented
19SD16×D97—28no—
20C2×D776C2212yescomplemented
21C2×#13095496C2212yescomplemented
22C2×#13095496C2212yescomplemented
23D8×D97—28no—
24C2×C8×D97C2212yes—
25#15690807C2211yescharacteristic, complemented
26#15690806—22no—
27#15690806C2212yescomplemented
28#15690799C2211yescharacteristic, complemented
29#15690798C2211yescharacteristic, complemented
30#15690797C2211yescharacteristic, complemented
31#15690796—22no—
32C22⋊C4×D97—22no—
33C42×D97C2211yescharacteristic
34#15690721C2211yescharacteristic, complemented
35#15690720C2212yescomplemented
36C2×SD16×C97C2212yescomplemented
37C2×D8×C97C2212yescomplemented
38#127×C97C2211yescharacteristic, complemented
39#124×C97C2211yescharacteristic, complemented
40#97×C97C2211yescharacteristic, complemented
41C23×D97—21no—
42C23×D97—22no—
43C2×Dic194—21no—
44C2×Dic194C2311yescharacteristic
45Q8×D97—24no—
46C2×D388—22no—
47C2×D388C2311yescharacteristic
48C2×D388C2312yes—
49C2×C22⋊D97—21no—
50C2×C22⋊D97—22no—
51C2×C22⋊D97—24no—
52D4×D97—24no—
53D4×D97—24no—
54D4×D97—24no—
55D4×D97—48no—
56C2×C4×D97—21no—
57C2×C4×D97D412yes—
58C2×C4×D97C2311yescharacteristic
59#13095500—24no—
60#13095499—24no—
61#13095498—24no—
62D776—24no—
63#13095496—24no—
64#13095496—24no—
65C2×#11715895C2312yes—
66C8×D97D418yescomplemented
67C22⋊Dic97—22no—
68#13095488—22no—
69#13095488—22no—
70C4×Dic97D412yes—
71C4×Dic97C2311yescharacteristic
72C2×Q8×C97C2311yescharacteristic
73C2×D4×C97—22no—
74C2×D4×C97C2311yescharacteristic
75C2×D4×C97C2312yes—
76SD16×C97—24no—
77D8×C97—24no—
78C2×C8×C97C2312yes—
79C22⋊C4×C97—22no—
80C42×C97C2311yescharacteristic
81C23×C97—21no—
82C23×C97—22no—
83C22×D97—21no—
84C22×D97—22no—
85C22×D97—42no—
86C22×D97—44no—
87C22×D97—48no—
88C22×D97—44no—
89C22×D97C2×D411yescharacteristic
90Q8×C97—22no—
91D4×C97—22no—
92D4×C97—22no—
93D4×C97—22no—
94D4×C97—44no—
95C2×C4×C97—21no—
96C2×C4×C97C2×D412yes—
97C2×C4×C97C2411yescharacteristic
98C22⋊D97—48no—
99C22⋊D97—42no—
100C22⋊D97—44no—
101C2×Dic97—21no—
102C2×Dic97C2×D412yes—
103C2×Dic97C2×D411yescharacteristic
104D388—22no—
105D388—22no—
106D388—22no—
107D388—44no—
108C4×D97—24no—
109C4×D97—42no—
110C4×D97C2×D414yes—
111Dic194—22no—
112Dic194—42no—
113C8×C97C2×D414yescomplemented
114#11715895C2×D414yescomplemented
115C22×C97—41no—
116C22×C97—42no—
117C22×C97—44no—
118C22×C97—42no—
119C22×C97C22×D411yescharacteristic
120C2×D97—41no—
121C2×D97—42no—
122C2×D97—44no—
123C2×D97—42no—
124C2×D97—88no—
125C2×D97—84no—
126C2×D97#12712yes—
127C2×D97AGL(1,ℤ/8)14yes—
128C4×C97—22no—
129C4×C97—41no—
130C4×C97C22×D412yes—
131Dic97—22no—
132Dic97—41no—
133Dic97#12712yes—
134C2×C97—82no—
135C2×C97—81no—
136C2×C97C2×#12711yescharacteristic
137C2×C97C2×AGL(1,ℤ/8)12yes—
138D97—82no—
139D97—81no—
140D97#49614yesdirect product factor
141C2×#496—971nomaximal, Sylow
142C97C2×#49611yescharacteristic, complemented, Sylow
143C22×SD16—972no—
144C22×D8—972no—
145C2×#127—971no—
146C2×#124—971no—
147#496—972no—
148#496—972no—
149#496—971no—
150#496—971no—
151#496—971no—
152#496D9711yescharacteristic, direct product factor
153C2×#97—971no—
154C22×Q8—971no—
155C22×D4—971no—
156C22×D4—972no—
157C22×D4—1942no—
158C2×SD16—972no—
159C2×SD16—972no—
160C2×SD16—972no—
161C2×SD16—1948no—
162C2×SD16C2×D9712yescomplemented
163C2×D8—972no—
164C2×D8—972no—
165C2×D8—972no—
166C2×D8—1948no—
167C2×D8C2×D9712yescomplemented
168C22×C8—972no—
169#127—972no—
170#127—971no—
171#127—1942no—
172#127C2×D9711yescharacteristic, complemented
173#124—971no—
174#124—971no—
175#124—971no—
176#124—1942no—
177#124C2×D9711yescharacteristic, complemented
178C2×C22⋊C4—1942no—
179C2×C42—971no—
180#97—972no—
181#97—971no—
182#97C2×D9711yescharacteristic, complemented
183C24—1941no—
184C24—1942no—
185C2×Q8—971no—
186C2×Q8—1941no—
187C2×Q8—1944no—
188C2×Q8C22×D9711yescharacteristic
189C2×D4—22no—
190C2×D4—971no—
191C2×D4—972no—
192C2×D4—3888no—
193C2×D4—1944no—
194C2×D4—1944no—
195C2×D4—1944no—
196C2×D4—1941no—
197C2×D4—1944no—
198C2×D4—1942no—
199C2×D4—1942no—
200C2×D4C22×D9711yescharacteristic
201C2×D4C22×D9712yes—
202C22×C4—971no—
203C22×C4—972no—
204C22×C4—1941no—
205SD16—24no—
206SD16—1944no—
207SD16—1944no—
208SD16—1944no—
209D8—24no—
210D8—1944no—
211D8—1944no—
212D8—1944no—
213C2×C8—972no—
214C2×C8—978no—
215C2×C8C22×D9712yes—
216C22⋊C4—22no—
217C22⋊C4—1942no—
218C22⋊C4—1942no—
219C22⋊C4—1942no—
220C42—972no—
221C42—971no—
222C42C22×D9711yescharacteristic
223C23—21no—
224C23—22no—
225C23—971no—
226C23—3888no—
227C23—3882no—
228C23—3884no—
229C23—3884no—
230C23—1941no—
231C23—1942no—
232Q8—22no—
233Q8—3882no—
234Q8—1942no—
235D4—22no—
236D4—22no—
237D4—22no—
238D4—44no—
239D4—3884no—
240D4—3882no—
241D4—3884no—
242D4—3888no—
243D4—1942no—
244D4—1942no—
245D4—1942no—
246C2×C4—21no—
247C2×C4—971no—
248C2×C4—974no—
249C2×C4—972no—
250C2×C4—3882no—
251C2×C4—1944no—
252C2×C4—1941no—
253C2×C4D4×D9712yes—
254C2×C4C23×D9711yescharacteristic
255C8—974no—
256C8D4×D9714yescomplemented
257C22—44no—
258C22—42no—
259C22—41no—
260C22—42no—
261C22—7768no—
262C22—7764no—
263C22—974no—
264C22—972no—
265C22—3884no—
266C22—3882no—
267C22—3881no—
268C22—3882no—
269C22C2×D4×D9711yescharacteristic
270C4—22no—
271C4—41no—
272C4—972no—
273C4—3881no—
274C4—1942no—
275C4C2×D4×D9712yes—
276C2—82no—
277C2—81no—
278C2—7762no—
279C2—7761no—
280C2—974no—
281C2#127×D9711yescharacteristic
282C2AGL(1,ℤ/8)×D9712yes—
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#496×C97C2
socleC22×C97C22×D4
hypercenter#496D97
hypocenterC97C2×#496
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #496

Complements

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

NH ≅ G/Ntype
1. C2×SD16×D97C2semidirect
2. C2×D8×D97C2semidirect
3. #127×D97C2semidirect
4. #124×D97C2semidirect
5. #20261125C2semidirect
6. #20261124C2semidirect
7. #20261123C2semidirect
8. #20261122C2semidirect
9. #20261121C2semidirect
10. #97×D97C2semidirect
11. #496×C97C2semidirect
16. C2×#13095500C22semidirect
17. C2×#13095499C22semidirect
18. C2×#13095498C22semidirect
20. C2×D776C22semidirect
21. C2×#13095496C22semidirect
22. C2×#13095496C22semidirect
25. #15690807C22semidirect
27. #15690806C22semidirect
28. #15690799C22semidirect
29. #15690798C22semidirect
30. #15690797C22semidirect
34. #15690721C22semidirect
35. #15690720C22semidirect
36. C2×SD16×C97C22semidirect
37. C2×D8×C97C22semidirect
38. #127×C97C22semidirect
39. #124×C97C22semidirect
40. #97×C97C22semidirect
66. C8×D97D4semidirect
113. C8×C97C2×D4semidirect
114. #11715895C2×D4semidirect
140. D97#496direct (N × H)
142. C97C2×#496semidirect
152. #496D97direct (N × H)
162. C2×SD16C2×D97semidirect
167. C2×D8C2×D97semidirect
172. #127C2×D97semidirect
177. #124C2×D97semidirect
182. #97C2×D97semidirect
256. C8D4×D97semidirect

Maximal subgroups

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

subgroupindex
C2×SD16×D972
C2×D8×D972
#127×D972
#124×D972
#202611252
#202611242
#202611232
#202611222
#202611212
#97×D972
#496×C972
C2×#49697