A Bayesian design is presented that does precision dose finding based on time to toxicity in a phase I clinical trial with two or more patient subgroups. The design, called Sub?TITE, makes sequentially adaptive subgroup?specific decisions while possibly combining subgroups that have similar estimated dose?toxicity curves. Decisions are based on posterior quantities computed under a logistic regression model for the probability of toxicity within a fixed follow?up period, as a function of dose and subgroup. Clustering priors are assumed for subgroup parameters, with latent subgroup combination variables included in the logistic model to allow different subgroups to be combined for dose finding if they are homogeneous. This framework can be used in trials where clinicians have identified patient subgroups but are not certain whether they will have different dose?toxicity curves. A simulation study shows that, when two or more subgroups are truly homogeneous but differ from other subgroups, the Sub?TITE design is substantially superior to either ignoring subgroups, running separate trials within all subgroups, or other proposed subgroup trials that borrow strength but do not cluster.