We present a multi-scale analysis of molecular hydrogen in a Milky Way-like simulated galaxy. Our census covers the gas content of the entire disc, to radial profiles and the Kennicutt-Schmidt relation, to a study of its molecular clouds, and finally down to a cell-by-cell analysis of the gas phases. Where observations are available we find agreement. A significant fraction of the H 2 gas is in low-density regions mixed with atomic hydrogen and would therefore be difficult to observe. We use the molecular addition to ramses-rt, an adaptive mesh refinement grid code with the hydrodynamics coupled to moment-based radiative transfer. Three resolutions of the same galaxy detail the effects it has on H 2 formation, with grid cells sized 97, 24, and 6.1 pc. Only the highest resolution yields gas densities high enough to host significant H 2 fractions, and resolution is therefore key to simulating H 2 . Apart our pieces of galactic analysis are disparate, but assembled they provide a cohesive portrait of H 2 in the interstellar medium. H 2 chemistry on the atomic scale is sufficient to generate its dynamics throughout an entire galaxy.