We address the electronic properties of quantum dots in the two-dimensional $alpha-mathcal{T}_3$ lattice when subjected to a perpendicular magnetic field. Implementing an infinite mass boundary condition, we first solve the eigenvalue problem for an isolated quantum dot in the low-energy, long-wavelength approximation where the system is described by an effective Dirac-like Hamiltonian that interpolates between the graphene (pseudospin 1/2) and Dice (pseudospin 1) limits. Results are compared to a full numerical (finite-mass) tight-binding lattice calculation. In a second step we analyse charge transport through a contacted $alpha-mathcal{T}_3$ quantum dot in a magnetic field by calculating the local density of states and the conductance within the kernel polynomial and Landauer-Buttiker approaches. Thereby the influence of a disordered environment is discussed as well.
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