Arranging numbers in a grid

In how many different ways can you arrange the numbers 1-9 in a 3X9 grid such that each number appears exactly once in each row: and each number appears exactly once in each of the left, middle, and right 3X3 grids? Here is the grid along with one possible arrangement:

2 7 1 3 5 9 6 4 8

4 3 8 6 7 2 1 5 9

5 6 9 1 4 8 2 3 7

If you start with an ordered grid like the following:

1 2 3 4 5 6 7 8 9

4 5 6

7 8 9

Then the total ways to arrange the numbers would be 9! for the left hand section. For the middle section would you have 9! - 3*3! and for the right section 9! - 3*6! ?

I can't seem to wrap my feeble mind around this problem.

Any hints would be greatly appreciated. Thanks.