The smallest solvable non-nilpotent group is [[symmetric group:S3|the symmetric group on three letters]]. This is [[centerless group|centerless]], so it cannot be nilpotent. On the other hand, it is clearly solvable, because its commutator subgroup is the alternating group on three letters, which is Abelian.
, any [[dihedral group]] whose order is not a power of 2, is solvable but not nilpotent. Also, for any prime <math>p</math>, the [[holomorph]] of the cyclic group of order <math>p</math> (i.e. its semidirect product with its automorphism group) is solvable, but ''not'' nilpotent.