What is the tightest lower bound currently known for the Carmichael function?

I imagine it must grow much more slowly than the Euler's totient function which according to here is bounded as

$$ \phi(n) \ge \frac{n}{e^{\gamma} \log \log n+ \frac{3}{\log \log n}} $$

But searching around on Google it's hard to find any literature on lower bounds of the carmichael function