  In this paper, we investigate extreme values of $\omega(E(\mathbb{F}_p))$,
where $E/\mathbb{Q}$ is an elliptic curve with complex multiplication and
$\omega$ is the number-of-distinct-prime-divisors function. For fixed $\gamma >
1$, we prove that \[ \#\{p \leq x : \omega(E(\mathbb{F}_p)) > \gamma\log\log
x\} = \frac{x}{(\log x)^{2 + \gamma\log\gamma - \gamma + o(1)}}. \] The same
result holds for the quantity $\#\{p \leq x : \omega(E(\mathbb{F}_p)) <
\gamma\log\log x\}$ when $0 < \gamma < 1$. The argument is worked out in detail
for the curve $E : y^2 = x^3 - x$, and we discuss how the method can be adapted
for other CM elliptic curves.
