It is well known that a finite group whose all proper subgroups are cyclic is either cyclic or direct product of quaternion group with cyclic group of odd order (am I correct?)
Question: What are the finite groups, whose proper quotients are cyclic?
[Here proper quotient of $G$ means the quotient of $G$ by non-trivial subgroup]
My (incomplete) answer is: if $G$ is a $p$-group then it should be cyclic or $C_p\times C_p$. In non-nilpotent group, some examples are $D_{2p}$ (dihedral groups of order $2p$, $p$ being prime).
Here is a list of such groups including those appearing in comments; I do not know whether they are classified.
Cyclic, dihedral of order $2p$ where $p$ is prime,
$S_n$ ($n\neq 4$),
simple groups,
$N\rtimes C_p$ where $N$ is non-abelian simple group and semi-direct product is not direct product..