  We classify four qubit states under SLOCC operations, that is, we classify
the orbits of the group $\mathrm{\mathop{SL}}(2,\mathbb{C})^4$ on the Hilbert
space $\mathcal{H}_4 = (\mathbb{C}^2)^{\otimes 4}$. We approach the
classification by realising this representation as a symmetric space of maximal
rank. We first describe general methods for classifying the orbits of such a
space. We then apply these methods to obtain the orbits in our special case,
resulting in a complete and irredundant classification of
$\mathrm{\mathop{SL}}(2,\mathbb{C})^4$-orbits on $\mathcal{H}_4$. It follows
that an element of $(\mathbb{C}^2)^{\otimes 4}$ is conjugate to an element of
precisely 87 classes of elements. Each of these classes either consists of one
element or of a parametrised family of elements, and the elements in the same
class all have equal stabiliser in $\mathrm{\mathop{SL}}(2,\mathbb{C})^4$. We
also present a complete and irredundant classification of elements and
stabilisers up to the action of ${\rm
Sym}_4\ltimes\mathrm{\mathop{SL}}(2,\mathbb{C})^4$ where ${\rm Sym}_4$ permutes
the four tensor factors of $(\mathbb{C}^2)^{\otimes 4}$.
