I have read in Wikipedia that all finite simple groups of Lie type have a (B, N) pair. What finite simple groups don't have a (B, N) pair? I presume that the answer is all prime-order cyclic groups, all alternating groups with order $60$ or more, the Tits group, and all sporadic groups.
Edit: I found out that $A_5$, $A_6$, and $A_8$ are all of Lie type, so they have a (B, N) pair.