The indexing you can do as long as you have the lattice parameters, i.e. you don't need the space groups. The interplanar distance is only a factor of (hkl) and the lattice parameters which you obviously have. If you observe serial, zonal, or general extinctions in your XRD pattern you can use them to predict or derive symmetry elements with translational components like screw axis and glide planes, or lattice centerings like A,B,C, I, R, and F.

A good approach is to start from a higher symmetric phase and go to lower symmetries by subgroup relationships. This does not work always, but a crystal structure often does not change their entire coordination, i.e. often only small shifts of a part of atoms are observed which change the lattice metric or even only the structure symmetry. if this simple approech does not work you need a crystal structure determination. You could also look for similar phases described by the same structure type (I guess in your case it is spinel, right? )