Let K be the least normal modal logic and BK its Belnapian version, which enriches K with ‘strong negation’. We carry out a systematic study of the lattice of logics containing BK based on:

— introducing the classes (or rather sublattices) of so-called explosive, complete and classical Belnapian modal logics;

— assigning to every normal modal logic three special conservative extensions in these classes;

— associating with every Belnapian modal logic its explosive, complete and classical counterparts.

We investigate the relationships between special extensions and counterparts, provide certain handy characterisations and suggest a useful decomposition of the lattice of logics containing BK.