Firstly, this must be an even number. This is because if its tens digit is odd, then the units digit must be $6$ (see # 4151). Therefore, the last two digits cannot be the same.

Because an even square must be a multiple of $4$ and cannot end with $8$, the last two digits can only be $44$. This means the last three digits can only be $444$.

However, $444$ is not a square but $1444=38^2$ is. Therefore the answer is $\boxed{1444}$.