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Numerical simulations constitute a precious tool for understanding the role of key parameters influencing the colloidal arrangement in suspensions, which is crucial for many applications. The present paper investigates numerically the role of hydrodynamic interactions on the aggregation processes in colloidal suspensions. Three simulation techniques are used: Brownian dynamics without hydrodynamic interactions, Brownian dynamics including some of the hydrodynamic interactions, using the Yamakawa-Rotne-Prager tensor, and stochastic rotation dynamics coupled with molecular dynamics. A system of monodisperse colloids strongly interacting through a generalized Lennard-Jones potential is studied for a colloid volume fraction ranging from 2.5 to 20%. Interestingly, effects of the hydrodynamic interactions are shown in the details of the aggregation processes. It is observed that the hydrodynamic interactions slow down the aggregation kinetics in the initial nucleation stage, while they speed up the next cluster coalescence stage. It is shown that the latter is due to an enhanced cluster diffusion in the simulations including hydrodynamic interactions. The higher the colloid volume fraction, the more pronounced the effects on the aggregation kinetics. It is also observed that hydrodynamic interactions slow down the reorganization kinetics. It turns out that the Brownian dynamics technique using the Yamakawa-Rotne-Prager tensor tends to overestimate the effects on cluster diffusion and cluster reorganization, even if it can be a method of choice for very dilute suspensions.