The finite simple groups taught in undergraduate or graduate courses are only up to $A_n$ or $\rm{PSL}_n$. Even many undergraduate and graduate texts do not consider simple groups beyond these two (one such graduate text I saw is Groups and Representations-Alperin, Bell).
What could be the other families of simple groups which can be introduced in undergraduate, assuming that the students know only Group Theory (up to Sylow theorems) and possibly Linear Algebra? And what is better, simple way of introducing it for undergraduates?