Yes. Note that Daniel Loughran's comment to David Speyer's answer to [this question][1] states that for any solvable group $G$, there is a Galois extension $L/\mathbb{Q}$ with all decomposition groups cyclic.

It's not hard to find a concrete example. If you let $L$ be the splitting field of $x^{4} - x^{3} - 7x^{2} + 2x + 9$, then $L/\mathbb{Q}$ is an $A_{4}$ extension ramified only at $163$. (This is one of the $A_{4}$ fields from Klueners and Malle's [excellent database][2].) There are four prime ideals, $\mathfrak{p}_{1}$, $\mathfrak{p}_{2}$, $\mathfrak{p}_{3}$ and $\mathfrak{p}_{4}$ above $163$ in $L$, and so $ef = 3$ for each such prime ideal. Therefore, the decomposition group for each such prime ideal has order $3$.


  [1]: https://mathoverflow.net/questions/240439/hasse-principle-for-rational-times-square
  [2]: http://galoisdb.math.upb.de/groups/view?deg=4&num=4