In this thesis, we consider a general construction of dihedral subgroups $D_n$, in the automorphism group of a complex finite-dimensional simple Lie algebra $g$. Our main results are based on Kac's classification results of finite order automorphisms on $g$ and a classic result on classification of $Z$-gradings on $g$. A dihedral subgroup $D_n$ in the automorphism group of $g$ can be defined by an order $n$ automorphism $sigma$ and an order 2 automorphism $r$ of $g$ satisfying the relation $rsigma = sigma^{-1}r$.
The key point to find the generators $(sigma , r)$ of the dihedral subgroups is to fix the order $n$ automorphism $sigma$ and extend an involution $r_0$ on the fixed point of $sigma$ to $r$ on the whole Lie algebra $g$. Finally, We give a complete list of classification of $(sigma , r)$ generating $D_3$-subgroups in the automorphism group of $g$.