* any [[dihedral group]] whose order is not a power of 2, is solvable but not nilpotent.
* for any prime <math>p</math>, the [[general affine group of degree one]] <math>GA(1,p)</math>, which can also be defined as 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.
* if <math>p</math> and <math>q</math> are primes such that <math>p</math> divides <math>q - 1</math>, there is a solvable non-nilpotent group of order <math>pq</math>. See [[classification of groups of order pq]]