  We determine all Nichols algebras of finite-dimensional Yetter-Drinfeld
modules over groups such that all its left coideal subalgebras in the category
of $\mathbb{N}_0$-graded comodules over the group algebra are generated in
degree one as an algebra. Here we confine ourselves to Yetter-Drinfeld modules
in which each group-homogeneous component is at most one-dimensional. We
present a strategy to extend left coideal subalgebras by adding a suitable
generator in degree two, three or four to a smaller left coideal subalgebra. We
also discuss some methods for the construction of left coideal subalgebras of a
Nichols algebra in the category of $\mathbb{N}_0$-graded $H$-comodules, where
$H$ is a Hopf algebra, that is not necessarily a group algebra.
