This paper models the current pandemic to analyze vaccination strategies in a setting with three age groups that differ with respect to their fatality rates. The model also accounts for heterogeneity in the transmission rates between and within these age groups. We compare the outcomes in terms of the total number of deceased, the total number of infected, the peak infection rate and the economic consequences. We find that fatalities are almost always minimized by first vaccinating the elderly, except when vaccination is slow and the general transmission rate is relatively low. In this case deaths are minimized by first vaccinating the middle-aged as this group is responsible for substantial spreading of the virus to the elderly. With regard to the other outcome variables it is always best to vaccinate the middle-aged group first. A trade-off may therefore emerge between reducing fatalities on the one hand and lowering the number of infected as well as maximizing the economic gains from vaccinations on the other hand.