Carbon nanotubes are widely used in a growing number of applications. Their interactions with biological materials, cell membranes in particular, is of interest in applications including drug delivery and for understanding the toxicity of carbon nanotubes. We use extensive molecular dynamics simulations with the MARTINI model to study the interactions of model nanotubes of different thickness, length, and patterns of chemical modification with model membranes. In addition, we characterize the interactions of small bundles of carbon nanotubes with membrane models. Short pristine carbon nanotubes readily insert into membranes and adopt an orientation parallel to the plane of the membrane in the center of the membrane. Larger aggregates and functionalized nanotubes exhibit a range of possible interactions. The distribution and orientation of carbon nanotubes can be controlled by functionalizing the nanotubes. Free energy calculations provide thermodynamic insight into the preferred orientations of different nanotubes and quantify structural defects in the lipid matrix.