In https://en.wikipedia.org/wiki/Cluster_decay - pointed out by @JonCuster one finds
$\lambda = \ln 2 /T_c = \nu S P$
Comparing with $\alpha$ decay, $S$ - formation probability for heavier fragments can be much lower than $\alpha$, because they have more complicated structure, while $\alpha$ is a very good cluster in nucleus in many cases (in Li, C, ...).
Another term is $P$ - penetrability, which translates to possibility to overcome a Coulomb barrier. What is the "penetrability factor"? - but from elsewhere you find this equation:
$P \sim \exp(-2\pi z_1 z_2 e^2/\hbar v)$
From this one can see that (if you consider Z=92 nucleus) $z_1z_2$ is 180 for $\alpha$ (2*90) and 516 for carbon (6*86). Then you must have higher velocity $v$ for a similar $P$, which means a lower fraction of the the wavefunction can participate. And exponential makes it even more important.
One is used to a fact, that the Coulomb barrier plays a role when particle is approaching the nucleus from outside, but it is actually similar when the particle resides inside, because it is low in a potential well created by some field of all particles.