Action on 8 Points Group Name (Label) \(|H^1(\mathbb{Q}, \mathbf{X}^\star(\mathbf{T}))|\) Sha_C

8T1

\(C_8\) (8,1)

1

1

8T2

\(C_4\times C_2\) (8,2)

2

2

8T3

\(C_2^3\) (8,5)

2

1

8T4

\(D_4\) (8,3)

2

1

8T5

\(Q_8\) (8,4)

2

4

8T6

\(D_8\) (16,7)

1

1

8T7

\(OD_{16}\) (16,6)

1

2

8T8

\(QD_{16}\) (16,8)

1

2

8T9

\(D_4\times C_2\) (16,11)

2

1

8T10

\(C_2^2:C_4\) (16,3)

2

1

8T11

\(Q_8:C_2\) (16,13)

2

2

8T12

\(\mathrm{SL}(2,3)\) (24,3)

2

1

8T13

\(A_4\times C_2\) (24,13)

2

1

8T15

\(Z_8:Z_8^\times\) (32,43)

1

2

8T16

\((C_8:C_2):C_2\) (32,7)

1

1

8T17

\(C_4\wr C_2\) (32,11)

1

1

8T18

\(C_2^2 \wr C_2\) (32,27)

2

1

8T19

\(C_2^3 : C_4\) (32,6)

2

1

8T20

\(C_2^3 : C_4\) (32,6)

1

1

8T21

\(C_2^3 : C_4\) (32,6)

1

1

8T22

\(C_2^3 : D_4\) (32,49)

1

1

8T23

\(\mathrm{GL(2,3)}\) (48,29)

1

1

8T24

\(S_4\times C_2\) (48,48)

2

1

8T26

\((C_4^2:C_2):C_2\) (64,134)

1

1

8T27

\(((C_8:C_2):C_2):C_2\) (64,32)

1

1

8T28

\((((C_4\times C_2):C_2):C_2):C_2\) (64,32)

1

1

8T29

\((((C_4\times C_2):C_2):C_2):C_2\) (64,138)

2

1

8T30

\((((C_4\times C_2):C_2):C_2):C_2\) (64,34)

1

1

8T31

\((((C_4\times C_2):C_2):C_2):C_2\) (64,138)

1

1

8T32

\(((C_2\times D_4):C_2):C_3\) (96,204)

2

1

8T35

\(C_2\wr C_2\wr C_2\) (128,928)

1

1

8T38

\(C_2\wr A_4\) (192,201)

1

1

8T39

\(C_2^3:S_4\) (192,1493)

1

1

8T40

\(Q_8:S_4\) (192,1494)

1

1

Note. Transitive subgroups which are not listed do not correspond to Galois groups of CM-fields.