TITLE: Example of an amenable finitely generated and presented group with a non-finitely generated subgroup
QUESTION [8 upvotes]: I'm looking for an example of a finitely presented and finitely generated amenable group, that has a subgroup which is not finitely generated.
The question is easy for finitely generated amenable group and an example is the lamp-lighter group $C_2\wr \mathbb{Z}$. 
An Abelian and finitely generated group has no such subgroups. There exists a bigger class of groups with this property?

REPLY [2 votes]: By the way, you may enjoy the fact, due to G. Baumslag, that a standard wreath product $W\wr G$ with $W\neq 1$ and $G$ infinite, is never finitely presented; see Gilbert Baumslag. Wreath products of finitely presented groups. Math. Z. 75 , 22-28, 1961.
For finite presentability of permutational wreath products, see a paper by Cornulier: http://www.normalesup.org/~cornulier/wrea_fp.pdf