Number of subgroups of an infinite group

Question

Is there an infinite group with only a finite number of subgroups?

Answer 1

No. An infinite group either contains $\mathbb Z$, which has infinitely many subgroups, or each element has finite order, but then the union $G = \bigcup_{g \in G} \langle g \rangle$ must be made of infinitely many subgroups.

Answer 2

Note that there are infinite groups with only a finite number of normal subgroups. For example the following infinite groups are simple.

The finitary alternating group $A(\lambda)$ for any infinite cardinal $\lambda$.
$PSL_n(K)$, with $K$ an infinite field and $n\geq 2$.