Small values of Carmichael's lambda function

Let $\lambda (n)$ be the exponent of the multiplicative group $(\mathbb{Z}\mathrm{/}n\mathbb{Z}{)}^{\times }$, and set $L(x,y)=\mathrm{\#}\{n\le x:\lambda (n)\le y\}$. We prove an upper bound for $\log \frac{L(x,y)}{x}$ valid for $\exp (({\log }_{2}x{)}^{1+\epsilon })\le y\le x\mathrm{/}\exp (({\log }_{2}x{)}^{1+\epsilon })$. Our bound is asymptotically sharp under a plausible hypothesis on powersmooth shifted primes. As an application, we obtain a new upper bound on the count of odd $n\le x$ for which the order of $2$ modulo $n$ is appreciably smaller than $x^{1\mathrm{/}2}$.