Finite Group Databasebeta

← #27156516

Subgroups — #27156516

Order 12416.

1464 conjugacy classes of subgroups; only the 545 normal subgroups are shown.

normal subgroups only

This group has 1464 conjugacy classes of subgroups — too many to draw. Only its 545 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#20261571C211yesmaximal, characteristic, complemented
2#20261546C218yesmaximal, complemented
3#20261538C218yesmaximal, complemented
4#20261533C216yesmaximal, complemented
5C2×#15690836C216yesmaximal, complemented
6#20261502C211yesmaximal, characteristic, complemented
7C2×H(5,𝔽2)×C97C211yesmaximal, characteristic, complemented
8#15690858—21no—
9#15690857C2214yescomplemented
10#15690856C2214yescomplemented
11#15690855—23no—
12C2×#13095511C2216yescomplemented
13#41×D97—24no—
14#15690841C2214yescomplemented
15#15690840C22112yescomplemented
16#15690836C22112yescomplemented
17#15690836C22124yescomplemented
18#15690835C22112yescomplemented
19#15690830C22124yescomplemented
20C2×#13095499C22112yescomplemented
21C2×#13095496C22112yescomplemented
22#15690817C2218yescomplemented
23C2×#13095495C2216yescomplemented
24H(5,𝔽2)×C97—24no—
25H(5,𝔽2)×C97C2218yescomplemented
26C2×#41×C97—21no—
27C2×#41×C97C2211yescharacteristic
28C2×#41×C97C2213yescomplemented
29C22×D4×C97—23no—
30C22×D4×C97C2213yescomplemented
31#13095512—24no—
32#13095511—26no—
33#13095511—26no—
34#13095511C2314yes—
35#13095511C23112yescomplemented
36#13095510—212no—
37C2×Dic194C2316yescomplemented
38Q8×D97—24no—
39C2×D388C2316yescomplemented
40C2×C22⋊D97—26no—
41D4×D97—212no—
42C2×C4×D97—26no—
43#13095501C2318yescomplemented
44#13095499C23124yescomplemented
45#13095498C2318yescomplemented
46#13095496C23124yescomplemented
47#13095495C2314yes—
48#13095495C23112yescomplemented
49C2×#11715895C23112yescomplemented
50C24×C97—23no—
51#41×C97—28no—
52#41×C97—26no—
53#41×C97D414yes—
54#41×C97C23112yes—
55#41×C97C2314yes—
56C2×Q8×C97D411yescharacteristic
57C2×Q8×C97C2311yescharacteristic
58C2×D4×C97—23no—
59C2×D4×C97—224no—
60C2×D4×C97—23no—
61C2×D4×C97—26no—
62C2×D4×C97D413yes—
63C2×D4×C97C23112yes—
64C2×D4×C97C2313yes—
65C22×C4×C97—23no—
66C22×C4×C97C2313yes—
67C23×C97—23no—
68C23×C97—212no—
69C23×C97—86no—
70C23×C97C2×D413yes—
71C22×D97—26no—
72Q8×C97—21no—
73Q8×C97C2×D414yes—
74Q8×C97C2412yes—
75D4×C97—212no—
76D4×C97—212no—
77D4×C97—23no—
78D4×C97C2×D4112yes—
79D4×C97C2416yes—
80C2×C4×C97—21no—
81C2×C4×C97—212no—
82C2×C4×C97C2×D413yes—
83C2×C4×C97C2×D416yes—
84C2×C4×C97C2411yescharacteristic
85C2×C4×C97C2416yes—
86C22⋊D97—212no—
87C22⋊D97—24no—
88C2×Dic97—26no—
89D388—26no—
90D388C2414yes—
91C4×D97—24no—
92C4×D97—212no—
93Dic194—26no—
94Dic194C2414yes—
95#11715895C2418yes—
96C22×C97—26no—
97C22×C97—43no—
98C22×C97—46no—
99C22×C97—812no—
100C22×C97C22×D416yes—
101C22×C97C22×D411yescharacteristic
102C2×D97—24no—
103C2×D97—86no—
104C4×C97—22no—
105C4×C97C22×D416yes—
106C4×C97C22×D411yescharacteristic
107C4×C97C2511yescharacteristic
108Dic97—24no—
109C2×C97—26no—
110C2×C97—21no—
111C2×C97—46no—
112C2×C97C23×D411yescharacteristic
113D97—44no—
114#3585—971nomaximal, Sylow
115C97#358511yescharacteristic, complemented, Sylow
116#585—971no—
117C2×H(5,𝔽2)D9711yescharacteristic, complemented
118#577—978no—
119#576—978no—
120#575—976no—
121C2×AGL(1,ℤ/8)—976no—
122#568—971no—
1232−1+4—974no—
1242−1+4—1941no—
125H(5,𝔽2)—24no—
126H(5,𝔽2)—974no—
127H(5,𝔽2)—1943no—
128H(5,𝔽2)C2×D9718yescomplemented
129C2×#41—21no—
130C2×#41—976no—
131C2×#41—1944no—
132C2×#41C2×D9711yescharacteristic
133C2×#41C2×D9713yescomplemented
134C22×D4—23no—
135C22×D4C2×D9713yescomplemented
136#137—9712no—
137#137—974no—
138AGL(1,ℤ/8)—9712no—
139AGL(1,ℤ/8)—9712no—
140AGL(1,ℤ/8)—9724no—
141#135—9724no—
142C2×SD16—9712no—
143C2×D8—9712no—
144#131—978no—
145C2×M16—976no—
146C24—23no—
147#41—28no—
148#41—26no—
149#41—974no—
150#41—9712no—
151#41—1944no—
152#41—19412no—
153#41—1946no—
154#41—1946no—
155#41C22⋊D9714yes—
156#41C22×D97112yes—
157#41C22×D9714yes—
158C2×Q8—976no—
159C2×Q8—1944no—
160C2×Q8C22⋊D9711yescharacteristic
161C2×Q8C22×D9711yescharacteristic
162C2×D4—23no—
163C2×D4—224no—
164C2×D4—23no—
165C2×D4—26no—
166C2×D4—976no—
167C2×D4—1946no—
168C2×D4—19412no—
169C2×D4C22⋊D9713yes—
170C2×D4C22×D97112yes—
171C2×D4C22×D9713yes—
172C22×C4—23no—
173C22×C4—1946no—
174C22×C4C22×D9713yes—
175Q16—978no—
176SD16—978no—
177SD16—9724no—
178D8—9724no—
179M16—974no—
180M16—9712no—
181C2×C8—9712no—
182C23—23no—
183C23—212no—
184C23—86no—
185C23—1946no—
186C23C2×C22⋊D9713yes—
187Q8—21no—
188Q8—974no—
189Q8—1946no—
190Q8C2×C22⋊D9714yes—
191Q8C23×D9712yes—
192D4—212no—
193D4—212no—
194D4—23no—
195D4—974no—
196D4—1944no—
197D4—19412no—
198D4—1946no—
199D4C2×C22⋊D97112yes—
200D4C23×D9716yes—
201C2×C4—21no—
202C2×C4—212no—
203C2×C4—1946no—
204C2×C4—1944no—
205C2×C4—19412no—
206C2×C4C2×C22⋊D9713yes—
207C2×C4C2×C22⋊D9716yes—
208C2×C4C23×D9716yes—
209C2×C4C23×D9711yescharacteristic
210C8—978no—
211C22—26no—
212C22—43no—
213C22—46no—
214C22—812no—
215C22—7766no—
216C22—1944no—
217C22C22×C22⋊D9711yescharacteristic
218C22C22×C22⋊D9716yes—
219C4—22no—
220C4—1944no—
221C4C22×C22⋊D9716yes—
222C4C22×C22⋊D9711yescharacteristic
223C4C24×D9711yescharacteristic
224C2—21no—
225C2—26no—
226C2—46no—
227C2—3884no—
228C2C23×C22⋊D9711yescharacteristic
subgroupG/H
centerC2C23×C22⋊D97
commutator, cosocleC4×C97C25
FrattiniC4C24×D97
FittingC2×H(5,𝔽2)×C97C2
socleC2×C97C23×D4
hypercenterC2×H(5,𝔽2)D97
hypocenterC97#3585
LCSG ▷ C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C2×#41 ◁ C2×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. #20261571C2semidirect
2. #20261546C2semidirect
3. #20261538C2semidirect
4. #20261533C2semidirect
5. C2×#15690836C2semidirect
6. #20261502C2semidirect
7. C2×H(5,𝔽2)×C97C2semidirect
9. #15690857C22semidirect
10. #15690856C22semidirect
12. C2×#13095511C22semidirect
14. #15690841C22semidirect
15. #15690840C22semidirect
16. #15690836C22semidirect
17. #15690836C22semidirect
18. #15690835C22semidirect
19. #15690830C22semidirect
20. C2×#13095499C22semidirect
21. C2×#13095496C22semidirect
22. #15690817C22semidirect
23. C2×#13095495C22semidirect
25. H(5,𝔽2)×C97C22semidirect
28. C2×#41×C97C22semidirect
30. C22×D4×C97C22semidirect
35. #13095511C23semidirect
37. C2×Dic194C23semidirect
39. C2×D388C23semidirect
43. #13095501C23semidirect
44. #13095499C23semidirect
45. #13095498C23semidirect
46. #13095496C23semidirect
48. #13095495C23semidirect
49. C2×#11715895C23semidirect
115. C97#3585semidirect
117. C2×H(5,𝔽2)D97semidirect
128. H(5,𝔽2)C2×D97semidirect
133. C2×#41C2×D97semidirect
135. C22×D4C2×D97semidirect

Maximal subgroups

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

subgroupindex
#202615712
#202615462
#202615382
#202615332
C2×#156908362
#202615022
C2×H(5,𝔽2)×C972
#358597