Sets, Subsets, Subgroups, Modulo 8

Question

This question has several questions within it pertaining to my title.
Consider the subset S={[0],[2],[3]} of the modulo 8 to be considered as the additive group modulo 8.
1) Determine the set S+S?
2)Is it a subset?
3)If H was a subgroup of modulo 8, what do you expect H+H to be?
4)Determine the subgroup generated by S, i.e., the smallest subgroup of modulo 8 contains S?
Here is my attempt to try to do it.Without doing the 1st question properly,I cant do the others.
My set modulo 8 is {[0],[1],[2],[3],[4],[5],[6],[7]}. Is using this set the correct way to determine S+S?

Answer 1

Assuming that $W=S+S$ means that for $W=\{ ((s+t)\bmod 8) \forall s,t\in S\}$ then $S+S=\{[0],[2], [3],[4],[5],[6]\}$.

We know it must be a subset (before even enumerating its members) because the additive group modulo $8$ is closed.

If $H$ is a subgroup, it must be closed: what does that say about $H+H$?

For the last question, consider that $3$ and $8$ are coprime.