<p>In computer experiments, it is common to encounter the situation of natural input grouping. This paper addresses such experiments by introducing a new type of orthogonal Latin hypercube designs, which exhibit superior orthogonality. For this type of designs, the within-group orthogonality exhibits a higher-order orthogonality property compared to the between-group orthogonality. This feature ensures that, in linear regression analyses, main effect estimates remain independent of other factors within the same group. Furthermore, to accommodate varying experimental sizes, we propose grouped nearly orthogonal Latin hypercube designs by relaxing the strict orthogonality within or between groups. Our construction methods are particularly advantageous for accommodating a large number of factors while offering flexibility in terms of groups and runs. We also provide a simulation study that demonstrates the advantage of our proposed designs over general orthogonal Latin hypercube designs, emphasizing the benefits of incorporating grouped structures. Additionally, we compare our method with several existing grouped designs, highlighting the differences across various design properties.</p>