We present a multiscale 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. A significant fraction of the H2 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 H2 formation, with grid cells sized 97, 24, and 6.1 pc. Only the highest resolution yields gas densities high enough to host significant H2 fractions, and resolution is therefore key to simulating H2. Our H2 content is not completely converged but we find general agreement with available observations. Apart our pieces of galactic analysis are disparate, but assembled, they provide a cohesive portrait of H2 in the interstellar medium. H2 chemistry on the atomic scale is sufficient to generate its dynamics throughout an entire galaxy.