Finite Group Databasebeta

← #27153502

Subgroups — #27153502

Order 12416.

464 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 464 conjugacy classes of subgroups — too many to draw. Only its 117 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
1#20261541C212yesmaximal, complemented
2#20261536C212yesmaximal, complemented
3#20261002C212yesmaximal, complemented
4#20260995C212yesmaximal, complemented
5#20260912C211yesmaximal, characteristic, complemented
6#20260907C211yesmaximal, characteristic, complemented
7#20260902C212yesmaximal, complemented
8#20260858C212yesmaximal, complemented
9#496×C97C211yesmaximal, characteristic, complemented
10#15690858C2211yescharacteristic, complemented
11#15690855C2211yescharacteristic
12#15690855C2212yescomplemented
13#15690841C2212yescomplemented
14#15690839—22no—
15#15690837—22no—
16#15690836C2212yes—
17#15690836C2212yescomplemented
18#15690836C2212yescomplemented
19#15690834—22no—
20#15690834—22no—
21#15690832C2212yes—
22#15690831—22no—
23#15690829—22no—
24SD16×D97—22no—
25D8×D97—22no—
26#15690816C2212yescomplemented
27#15690816C2212yescomplemented
28#15690746C2212yescomplemented
29#15690738C2212yescomplemented
30#15690738C2212yescomplemented
31#15690729C2211yescharacteristic, complemented
32#15690727C2212yescomplemented
33#15690727C2211yescharacteristic, complemented
34C2×SD16×C97C2212yescomplemented
35C2×D8×C97C2212yescomplemented
36#127×C97C2211yescharacteristic, complemented
37#124×C97C2211yescharacteristic, complemented
38#97×C97C2211yescharacteristic, complemented
39#13095512—22no—
40#13095511—41no—
41#13095511C2312yes—
42#13095511C2312yes—
43#13095510—22no—
44#13095510—22no—
45#13095510—22no—
46Q8×D97—22no—
47C2×C22⋊D97—41no—
48C2×C22⋊D97—42no—
49D4×D97—22no—
50D4×D97—22no—
51D4×D97—22no—
52Dic388D412yescomplemented
53#13095501—22no—
54#13095500D412yescomplemented
55#13095500D412yescomplemented
56#13095499—22no—
57#13095499—22no—
58#13095499—22no—
59#13095498—22no—
60D776D412yescomplemented
61#13095496—22no—
62#13095496—22no—
63#13095496—22no—
64#13095495C2312yes—
65#13095495C2312yes—
66#13095494—22no—
67#13095494—22no—
68C8×D97—22no—
69C8×D97—22no—
70C2×Q8×C97C2311yescharacteristic
71C2×D4×C97—41no—
72C2×D4×C97C2312yes—
73C2×D4×C97C2311yescharacteristic
74SD16×C97—22no—
75SD16×C97—22no—
76D8×C97—22no—
77D8×C97—22no—
78C2×C8×C97C2312yes—
79C22⋊C4×C97—41no—
80C42×C97C2311yescharacteristic
81C23×C97—21no—
82C23×C97—22no—
83C22×D97—42no—
84C22×D97—41no—
85Q8×C97—22no—
86D4×C97—22no—
87D4×C97—22no—
88D4×C97—22no—
89D4×C97—82no—
90C2×C4×C97—21no—
91C2×C4×C97—21no—
92C2×C4×C97C2411yescharacteristic
93C22⋊D97—22no—
94C22⋊D97—22no—
95C22⋊D97—42no—
96C22⋊D97—42no—
97C22⋊D97—42no—
98C2×Dic97—42no—
99C2×Dic97—41no—
100D388—41no—
101D388C2×D412yes—
102D388C2×D412yes—
103C4×D97—22no—
104C4×D97—22no—
105C4×D97—42no—
106Dic194—41no—
107Dic194C2×D412yes—
108Dic194C2×D412yes—
109C8×C97C2×D412yes—
110C8×C97C2×D412yes—
111#11715895—22no—
112#11715895—22no—
113C22×C97—41no—
114C22×C97—42no—
115C22×C97—82no—
116C22×C97—81no—
117C22×C97H(5,𝔽2)11yescharacteristic
118C2×D97—22no—
119C2×D97—22no—
120C2×D97—162no—
121C2×D97—161no—
122C4×C97—41no—
123C4×C97—41no—
124C4×C97C22×D412yes—
125Dic97—22no—
126Dic97—22no—
127C2×C97—21no—
128C2×C97—82no—
129C2×C97—81no—
130C2×C97D4211yescharacteristic
131D97—42no—
132D97—42no—
133#3288—971nomaximal, Sylow
134C97#328811yescharacteristic, complemented, Sylow
135#577—972no—
136#576—972no—
137#496D9711yescharacteristic, complemented
138#472—972no—
139#471—972no—
140#455—971no—
141#454—971no—
142#453—972no—
143#444—972no—
1442−1+4—971no—
145H(5,𝔽2)—971no—
146H(5,𝔽2)—972no—
147#137—972no—
148#137—1942no—
149AGL(1,ℤ/8)—972no—
150AGL(1,ℤ/8)—972no—
151AGL(1,ℤ/8)—972no—
152AGL(1,ℤ/8)—1942no—
153AGL(1,ℤ/8)—1942no—
154AGL(1,ℤ/8)—1942no—
155#135—972no—
156#135—1942no—
157#135—1942no—
158C2×SD16—1942no—
159C2×SD16C2×D9712yescomplemented
160C2×D8—1942no—
161C2×D8C2×D9712yescomplemented
162#131—972no—
163#131—972no—
164#127C2×D9711yescharacteristic, complemented
165#124C2×D9711yescharacteristic, complemented
166#108—972no—
167#104—972no—
168#104—972no—
169#101—971no—
170#100—972no—
171#100—971no—
172#97C2×D9711yescharacteristic, complemented
173#41—972no—
174#41—972no—
175#41—3881no—
176#41—1942no—
177#41—1942no—
178#41—1942no—
179#41—1942no—
180C2×Q8—1942no—
181C2×Q8C22×D9711yescharacteristic
182C2×D4—41no—
183C2×D4—3881no—
184C2×D4—3882no—
185C2×D4—1942no—
186C2×D4—1942no—
187C2×D4—1942no—
188C2×D4C22×D9711yescharacteristic
189C2×D4C22×D9712yes—
190Q16—972no—
191Q16—1942no—
192SD16—22no—
193SD16—22no—
194SD16—972no—
195SD16—972no—
196SD16—1942no—
197SD16—1942no—
198SD16—1942no—
199SD16—1942no—
200D8—22no—
201D8—22no—
202D8—972no—
203D8—1942no—
204D8—1942no—
205D8—1942no—
206M16—972no—
207M16—972no—
208M16—1942no—
209M16—1942no—
210C2×C8—1942no—
211C2×C8—1942no—
212C2×C8C22×D9712yes—
213C22⋊C4—41no—
214C42C22×D9711yescharacteristic
215C23—21no—
216C23—22no—
217C23—3882no—
218C23—3881no—
219Q8—22no—
220Q8—972no—
221Q8—972no—
222Q8—3881no—
223D4—22no—
224D4—22no—
225D4—22no—
226D4—82no—
227D4—972no—
228D4—972no—
229D4—3881no—
230D4—3882no—
231D4—3882no—
232D4—3882no—
233D4—1942no—
234D4—1942no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—3882no—
238C2×C4—3881no—
239C2×C4—3882no—
240C2×C4—1942no—
241C2×C4—1942no—
242C2×C4C23×D9711yescharacteristic
243C8—1942no—
244C8—1942no—
245C8D4×D9712yes—
246C8D4×D9712yes—
247C22—41no—
248C22—42no—
249C22—82no—
250C22—81no—
251C22—15521no—
252C22—15522no—
253C22—1942no—
254C22—1942no—
255C22#1569085511yescharacteristic
256C4—41no—
257C4—41no—
258C4—1942no—
259C4—1942no—
260C4C2×D4×D9712yes—
261C2—21no—
262C2—82no—
263C2—81no—
264C2—3882no—
265C2—3882no—
266C2#2026137011yescharacteristic
subgroupG/H
centerC2#20261370
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#496×C97C2
socleC2×C97D42
hypercenter#496D97
hypocenterC97#3288
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C2×C4 ◁ #496

Complements

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

NH ≅ G/Ntype
1. #20261541C2semidirect
2. #20261536C2semidirect
3. #20261002C2semidirect
4. #20260995C2semidirect
5. #20260912C2semidirect
6. #20260907C2semidirect
7. #20260902C2semidirect
8. #20260858C2semidirect
9. #496×C97C2semidirect
10. #15690858C22semidirect
12. #15690855C22semidirect
13. #15690841C22semidirect
17. #15690836C22semidirect
18. #15690836C22semidirect
26. #15690816C22semidirect
27. #15690816C22semidirect
28. #15690746C22semidirect
29. #15690738C22semidirect
30. #15690738C22semidirect
31. #15690729C22semidirect
32. #15690727C22semidirect
33. #15690727C22semidirect
34. C2×SD16×C97C22semidirect
35. C2×D8×C97C22semidirect
36. #127×C97C22semidirect
37. #124×C97C22semidirect
38. #97×C97C22semidirect
52. Dic388D4semidirect
54. #13095500D4semidirect
55. #13095500D4semidirect
60. D776D4semidirect
134. C97#3288semidirect
137. #496D97semidirect
159. C2×SD16C2×D97semidirect
161. C2×D8C2×D97semidirect
164. #127C2×D97semidirect
165. #124C2×D97semidirect
172. #97C2×D97semidirect

Maximal subgroups

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

subgroupindex
#202615412
#202615362
#202610022
#202609952
#202609122
#202609072
#202609022
#202608582
#496×C972
#328897