The Lie group analysis of the two-dimensional, time-independent, isotropic quantum harmonic oscillator is performed. Firstly, a simplified version of the equation called the base case equation is considered and its symmetry Lie algebra was found to be isomorphic to the Euclidean Lie algebra. Thus the symmetries that the base case equation possessed were rotations about the origin and translations in the x- and y-directions. Secondly, a new basis for the Lie algebra of the 2D harmonic oscillator was found and the equation was shown to possess rotational symmetries about the origin. Next, the base case equation was written in complex variables and its symmetries were used to obtain two group invariant solutions which were in turn used to form a general solution. Lastly, the two-dimensional harmonic oscillator was converted to complex form and its rotational symmetry was used to construct a general solution.