Example 12 Solve the high-order congruence equation
$$x^{6} \equiv 11(\bmod 19)$$
From Theorem 15 (3), the above equation can be transformed into an equivalent congruence equation in terms of indices:
$$6 \operatorname{ind}_{g} x \equiv \operatorname{ind}_{9} 11(\bmod \varphi(19))$$

By the table, we can take \( g \equiv 2(\bmod 19) \), and from the table we find
$$\text { ind }_{g} 11=12 \text {, }$$

Substituting this into the congruence equation, we get
$$6 \text { ind }_{g} x \equiv 12(\bmod 18)$$

Eliminating the common factor 6, we obtain
$$\operatorname{ind}_{g} x \equiv 2(\bmod 3)$$

Thus, there are six solutions:
$$\text { ind }_{g} x=2,5,8,11,14,17$$

Referring to the index table again, the corresponding six solutions are
$$x \equiv 4,13,9,15,6,10(\bmod 19)$$