Up to isomorphism there are fourteen groups of order \(16\). The five abelian ones are \[ \ZZ/16\ZZ,\quad(\ZZ/4\ZZ)^2,\quad\ZZ/8\ZZ\times\ZZ/2\ZZ,\quad\ZZ/4\ZZ\times(\ZZ/2\ZZ)^2,\quad(\ZZ/2\ZZ)^4, \] and the nine nonabelian ones are
\((\ZZ/4\ZZ\times\ZZ/2\ZZ)\rtimes\ZZ/2\ZZ=\gens{a,b,c\suchthat a^4=b^2=c^2=1,\ ab=ba,\ cac^{-1}=ab,\ cbc^{-1}=b}\),
\(\ZZ/4\ZZ\rtimes\ZZ/4\ZZ\), with a generator of the second factor acting by inversion,
the modular group \(\ZZ/8\ZZ\rtimes\ZZ/2\ZZ=\gens{r,s\suchthat r^8=s^2=1,\ srs^{-1}=r^5}\),
the dihedral group \(D_8\) of order \(16\),
\(D_4\times\ZZ/2\ZZ\), with \(D_4\) the dihedral group of order \(8\),
\(Q_8\times\ZZ/2\ZZ\), with \(Q_8\) the quaternion group,
the generalized quaternion group \(Q_{16}=\gens{a,b,c\suchthat a^4=b^2=c^2=abc}\),
the semidihedral group \(QD_{16}=\gens{r,s\suchthat r^8=s^2=1,\ srs^{-1}=r^3}\),
the Pauli group \((\ZZ/4\ZZ\times\ZZ/2\ZZ)\rtimes\ZZ/2\ZZ=\gens{a,b,c\suchthat a^4=b^2=c^2=1,\ ab=ba,\ cac^{-1}=a,\ cbc^{-1}=a^2b}\), generated by the Pauli matrices.
The groups in items 1 and 9 are both semidirect products \((\ZZ/4\ZZ\times\ZZ/2\ZZ)\rtimes\ZZ/2\ZZ\) and are not isomorphic: the center of the group in item 1 is \(\gens{a^2,b}\cong(\ZZ/2\ZZ)^2\), and the center of the Pauli group is \(\gens{a}\cong\ZZ/4\ZZ\).