We explore the relation between dust and several fundamental properties of simulated galaxies using the Dusty SAGE semi-analytic model. In addition to tracing the standard galaxy properties, Dusty SAGE also tracks cold dust mass in the interstellar medium (ISM), hot dust mass in the halo and dust mass ejected by feedback activity. Based on their ISM dust content, we divide our galaxies into two categories: ISM dust-poor and ISM dust-rich. We split the ISM dust-poor group into two subgroups: halo dust-rich and dust-poor (the latter contains galaxies that lack dust in both the ISM and halo). Halo dust-rich galaxies have high outflow rates of heated gas and dust and are more massive. We divide ISM dust-rich galaxies based on their specific star formation rate (sSFR) into star-forming and quenched subgroups. At redshift z=0, we find that ISM dust-rich galaxies have a relatively high sSFR, low bulge-to-total (BTT) mass ratio, and high gas metallicity. The high sSFR of ISM dust-rich galaxies allows them to produce dust in the stellar ejecta. Their metal-rich ISM enables dust growth via grain accretion. The opposite is seen in the ISM dust-poor group. Furthermore, ISM dust-rich galaxies are typically late-types, while ISM dust-poor galaxies resemble the early-type population, and we show how their ISM content evolves from being dust-rich to dust-poor. Finally, we investigate dust production from z=3 to z=0 and find that all groups evolve similarly, except for the quenched ISM dust-rich group.