It is well-known that log terminal singularities are rational, but log canonical
singularities are not. On the other hand, rational singularities are not necessarily
$\mathbb Q$-Gorenstein, so there are rational singularities that are not even log
canonical (and hence not log terminal). On the third hand (if someone has one),
rational Gorenstein singularities are canonical and hence log terminal and hence log
canonical.


So that leads to the question: Is there a rational singularity that is $\mathbb
Q$-Gorenstein, but not log canonical?



Rational, but not log canonical singularity
Actually, if you look at my [answer](https://mathoverflow.net/questions/123286/rational-and-log-canonical-singularity-that-is-not-log-terminal/123287#123287) to your previous [question](https://mathoverflow.net/questions/123286/rational-and-log-canonical-singularity-that-is-not-log-terminal), the same computation gives you an example for this as well. The example there is $\mathbb Q$-Gorenstein and if you choose the $n\_i$ such that the intersections matrix is negative definite and 
$$
\frac 1{n\_1}+\frac 1{n\_2}+\frac 1{n\_3}< 1
$$
holds, then you get a rational $\mathbb Q$-Gorenstein singularity that is not log canonical. 
I don't feel like computing determinants, but I am pretty certain that if you choose the $n\_i$ large enough then they will work. 


