Subgroup-discovery methods allow users to obtain simple descriptions of interesting regions in a dataset. Using constraints in subgroup discovery can enhance interpretability even further. In this article, we focus on two types of constraints: First, we limit the number of features used in subgroup descriptions, making the latter sparse. Second, we propose the novel optimization problem of finding alternative subgroup descriptions, which cover a similar set of data objects as a given subgroup but use different features. We describe how to integrate both constraint types into heuristic subgroup-discovery methods. Further, we propose a novel Satisfiability Modulo Theories (SMT) formulation of subgroup discovery as a white-box optimization problem, which allows solver-based search for subgroups and is open to a variety of constraint types. Additionally, we prove that both constraint types lead to an NP-hard optimization problem. Finally, we employ 27 binary-classification datasets to compare heuristic and solver-based search for unconstrained and constrained subgroup discovery. We observe that heuristic search methods often yield high-quality subgroups within a short runtime, also in scenarios with constraints.