No. This follows from a beautiful theorem of B.H. Neumann:
Let $G$ be a group. If $\{x_iH_i\}_{i=1}^n$ is a covering of $G$ by cosets of proper subgroups, then $n \geq \min_{i} [G:H_i]$.
Explicitly, this is Lemma 4.1 in
http://www.math.uga.edu/~pete/Neumann54.pdf
As Neumann remarks, the identity $gH = (g H g^{-1}) g$ shows that it is no loss of generality to restrict to coverings by left cosets.
