S-boxes (substitution boxes) are essential nonlinear elements that guarantee confusion in symmetric cryptosystems, but it is still difficult to come up with structures that have a good algebraic and cryptographic structure. This paper is an attempt to suggest a hybrid construction approach characterized by a combination of finite field GF(2n) transformations and permutation group techniques to create secure and efficient S-boxes. The method is a combination of algebraic inversion, affine mapping and permutation cycles to maximize nonlinearity, objectivity and resistance to differential and linear cryptanalysis. The proposed construction of hybrid S-box uses finite-field inversion, affine transformations and permutation-group operations to enhance the cryptographic substitution properties. The results reported here show a nonlinearity of 114, while AES S-box and traditional S-box have values of 112 and 96, respectively, which represent 1.79% and 18.75% improvement, respectively. The proposed construction meets the AES baseline by +1.4 and +2.4 percentage points on both the Strict Avalanche Criterion (SAC) and Bit Independence Criterion (BIC) respectively. The reported computational time is 11.1 ms which is 11.90% faster than traditional S-box but 7.77% slower than AES. These comparisons suggest that these characteristics are reported to be better with little to be gained in terms of computation.