8. Fill the first eight positive integers into a $2 \times 4$ table, with one number per cell, such that the four numbers in each row increase from left to right and the two numbers in each column increase from bottom to top. The number of different ways to fill the table is $\qquad$ .
8. 14 .

Obviously, 1 can only be placed in the lower left corner, and 8 can only be placed in the upper right corner.

Noting that, 2 is in the adjacent cell to 1, and 7 is in the adjacent cell to 8.
Therefore, we can calculate in the following four cases.
(1) As shown in Table 1, the positions of 3 and 6 are determined, so there are $\mathrm{C}_{2}^{1}=2$ ways;
(2) As shown in Table 2, the position of 3 is determined, so there are $\mathrm{C}_{3}^{1}=3$ ways;
(3) As shown in Table 3, the position of 6 is determined, so there are $\mathrm{C}_{3}^{1}=3$ ways;
(4) As shown in Table 4, there are $\mathrm{C}_{4}^{2}=6$ ways.

In summary, there are $2+3+3+6=14$ ways.