Linear Transformations · Superchallenge problem

Let $T:\mathbb{R}^4\to\mathbb{R}^4$ be $T(x_1, x_2, x_3, x_4) = (x_1 + x_2,\ x_2,\ 2x_3 + x_4,\ 2x_4)$. How many subspaces $W$ of $\mathbb{R}^4$ (counting $\{\mathbf 0\}$ and $\mathbb{R}^4$) are invariant under $T$, i.e. satisfy $T(W)\subseteq W$?