The CFSG (classification of finite simple groups) yields L: The finite groups of Lie type,and S: the non-Lie groups = 26 sporadic simples. We do not know how natural this taxonomy is.
One approach is that of (categorical) 2-groups. Another is that of BIRS Banff 12frg158 whichis an attempt to tame the sporadics using integrable systems, symplectic geometry, characteristic classes, and mathematical physics. This may lead to flourishing of new interconnections between many fields.