1. Classify the elements of $S$ modulo $3$.

We have $S=\{1,2,3,4,5,6,9\}.$ Their residues modulo $3$ are:

• $\{1,4\}\equiv 1 \pmod 3$
• $\{2,5\}\equiv 2 \pmod 3$
• $\{3,6,9\}\equiv 0 \pmod 3$

So the set contains:

• $2$ elements of type $1 \pmod 3$
• $2$ elements of type $2 \pmod 3$
• $3$ elements of type $0 \pmod 3$

2. Total number of non-empty subsets of $S$.

Since $|S|=7$, total subsets are $2^7=128.$ Hence total non-empty subsets are $128-1=127.$

We must exclude those non-empty subsets whose sum is divisible by $3$.

3. Count subsets whose sum is a multiple of $3$.

Let:

• $a$ = number of chosen elements from $\{1,4\}$
• $b$ = number of chosen elements from $\{2,5\}$
• $c$ = number of chosen elements from $\{3,6,9\}$

Then the subset sum modulo $3$ is determined by $a+2b \pmod 3,$ because elements from $\{3,6,9\}$ contribute $0 \pmod 3$.

Now count choices from each group according to residue contribution.

From $\{1,4\}$

Possible selections:

• choose $0$ elements: $\binom20=1$ way, contribution $0$
• choose $1$ element: $\binom21=2$ ways, contribution $1$
• choose $2$ elements: $\binom22=1$ way, contribution $2$

So counts by contribution mod $3$ are: $N_1(0)=1,\quad N_1(1)=2,\quad N_1(2)=1.$

From $\{2,5\}$

Possible selections:

• choose $0$ elements: $1$ way, contribution $0$
• choose $1$ element: $2$ ways, contribution $2$
• choose $2$ elements: $1$ way, contribution $1$

So counts by contribution mod $3$ are: $N_2(0)=1,\quad N_2(1)=1,\quad N_2(2)=2.$

From $\{3,6,9\}$

Any subset contributes $0 \pmod 3$. Number of choices: $2^3=8.$

4. Count selections from first two groups with total contribution $0 \pmod 3$.

We need residue pairs adding to $0 \pmod 3$:

• $(0,0)$
• $(1,2)$
• $(2,1)$

Hence number of such selections is $N_1(0)N_2(0)+N_1(1)N_2(2)+N_1(2)N_2(1)$ $=1\cdot 1+2\cdot 2+1\cdot 1=1+4+1=6.$

For each such selection, we can choose any subset of $\{3,6,9\}$ in $8$ ways.

Thus total subsets with sum divisible by $3$ are $6\times 8=48.$ This includes the empty subset.

So non-empty subsets with sum divisible by $3$ are $48-1=47.$

5. Required count.

We want non-empty subsets whose sum is not divisible by $3$: $127-47=80.$

Therefore, $\boxed{80}.$

6. Comparison with stored answer.

Stored correct answer = $80$.

Our derived answer also equals $80$, so it agrees.