How do I find the degree of the splitting field?

Question

I'm trying to find the degree of the splitting field of the polynomial $p(x) = x^{8}+5x^{4}-14$ over $\mathbb{Q}$. Here are my steps: Factoring p(x), we have $p(x) = (x^{4}-2)(x^{4}+7)$. Setting p(x) equal to $0$, $x=\pm\sqrt[4]{2}, \pm\sqrt[4]{-7}$. Letting $\zeta_{4}^{k} = e^{2ki\pi/4}, x=\pm\sqrt[4]{2}, \pm\sqrt[4]{-7}, \zeta_{4}(\pm\sqrt[4]{2}), \zeta_{4}(\pm\sqrt[4]{-7}), \zeta_{4}^{3}(\pm\sqrt[4]{2}), \zeta_{4}^{3}(\pm\sqrt[4]{-7})$. Then our splitting field of $p(x)$ is $\mathbb{Q}(\sqrt[4]{2}, \sqrt[4]{-7}, \zeta_{4}, \zeta_{4}^{3})$. Does this mean that the degree of the splitting field over $\mathbb{Q}$ is equal to $4+4+2+2=12$? This doesn't seem right.