Expected Value, Linearity & Advanced Probability · Superchallenge problem

Each of the $8$ vertices of a cube is coloured red, green or blue, independently and uniformly at random. An edge of the cube is monochromatic if its two endpoints have the same colour. Let $X$ be the number of monochromatic edges among the $12$ edges. Find the variance of $X$. Express your answer as a common fraction.