The question is a bit broad. Generating secure elliptic curves is highly non-trivial, see for example this question, which contains references to this and this paper.
Considering a black box group, the fact that it has order $6\cdot q$ does not break the DLP. What happens is that the DLP can easily be reduced to a group of order $q$ by eliminating the cofactor 6. So being a multiple of 6 does not break DLP, nor does it help.
However, an elliptic-curve group is of course not black box. In some particular cases we can use properties that reduce the ECDLP to simpler DLP's, for example in the case of supersingular or anomalous curves. These are criteria which have to be taken into consideration when choosing the curve, to make sure the corresponding ECDLP is hard.
However, cryptographic protocols rely on assumptions which are similar, but not exactly the same as the ECDLP. For example, the Diffie-Hellman assumption does not reduce to ECDLP. Therefore, we also have to choose curves in such a way that attacks on our particular protocol are avoided. There are many attacks which target an elliptic-curve-based protocols by using properties of the underlying curve. Important examples are small-subgroup attacks (very relevant here!) and twist attacks. There are many more examples.
In conclusion, having order $h\cdot q$ for a large prime $q$ and some integer $h$ is necessary to build secure elliptic-curve-based protocols. It is however not sufficient. Obtaining a secure elliptic curve of order $6\cdot q$ is possible, but you'll want to take many more properties into account before deciding to use it. This is very non-trivial, so generate your own curves with care.