If all Subgroups are Cyclic, is group Cylic? [duplicate]

I am having difficulty seeing if it is the case that if all subgroups of a group are cyclic, that the group itself is cyclic.

The group itself is a subgroup, so that it is cyclic as well, because all subgroups are cyclic. Did you maybe mean that all strict subgroups are cyclic ? In this case, it is wrong to conclude that the group itself is cyclic : for instance $G = \left\{ \frac{x}{2^d}\;|\;x\in\mathbf{Z}, d\in\mathbf{n}\right\}$ is not cyclic, and has only cyclic strict subgroups. (I am giving this example, as all given examples were finite groups.)

If you mean proper subgroups, then No! consider $G = \mathbf{Z}_2 \times \mathbf{Z}_2$.