No pattern this time

The given figure can be thought of as 》■《.
In the lower half of 》there are 36 quadrilaterals. So, in 》and《, there are a total of (4×36 = 144) quadrilaterals.

Now, come to ■. There are 144 quadrilaterals in this part.

So, the total number of quadrilaterals = $144 + 144 = \boxed{288}$ .

${4 \choose 2}\times{4 \choose 2} = 36$ is number of quadrilateral in one of the 4 parallelogram in the 4 corners.

${7 \choose 2}\times{4 \choose 2} = 126$ is number of rectangles in the central rectangle.

hence total is $126 + 36\times4 = \boxed{270}$