classical mechanics - Stiffness tensor

Let's have a stiffness tensor:

$$ a^{ijkl}: a^{ijkl} = a^{jikl} = a^{klij} = a^{ijlk}. $$

It has a 21 independent components for an anisotropic body.

How does body symmetry (cubic, hexagonal etc.) change the number of independent components of the tensor? For example, for cubiс symmetry it has three components. How to explain it?

Update.

Is the explanation a simple realization of idea $$ a_{ijkl}' = \beta_{im}a^{m}\beta_{jt}a^{t}\beta_{k f}a^{f}\beta_{ld}a^{d} = a_{ijkl}, $$ where $\beta_{\alpha \beta}$ is a components of a matrix $\beta$ for rotation around z-, x-, y-axis at the same time?