Which subproblems of 3SAT are still NP complete? The $\mathsf{NP}$-complete problems satisfy the property that if any of them is in $\mathsf{P}$, then all other $\mathsf{NP}$-complete problems are in $\mathsf{P}$.
The problem $3$-SAT is $\mathsf{NP}$-complete, as much as every $k$-SAT with $k\geq 3$.
(On the other hand, $2$-SAT is known to be in $\mathsf{P}$).

Obviously, the problem $3$-SAT can be seen as a subset of $k$-SAT for $k\geq 3$ and, in general, all of them are a subclass of SAT.

However my question goes in the opposite direction: which are some smaller subclasses of $3$-SAT that are still $\mathsf{NP}$-complete?

I am interested in studying the $\mathsf{P} = \mathsf{NP}$ conjecture by looking just into smaller class of $3$-SAT problems that are still acceptable.
I asked a similar question on cs.stackexchange.com, perhaps Juho's answer can help you; it contains references to: MONOTONE NAE-3SAT, MONOTONE 1-in-3-SAT, PLANAR 3SAT, k-COLOURABLE MONOTONE NAE-3SAT.

You can find other variants also in this cstheory question: 4-BOUNDED PLANAR 3-CONNECTED 3SAT, POSITIVE PLANAR 1-in-3 SAT.