The study of p-adic Lie groups and their representations is a central piece of the p-adic Langlands program.  One tool which is used to study these is the notion of a saturated pro-p group, and the famous result of Lazard which states that every p-adic Lie group contains an open saturable subgroup.  In this talk, we will demonstrate a family of open saturated subgroups of G(F) for G a reductive group over a p-adic field F, which is indexed by the semisimple Bruhat-Tits building of G, given a mild assumption on G.  We will then review some group-theoretic consequences of this result.