A device for the shock-absorbing attachment of two elements to be assembled, the attachment device including: an inner part that is tubular along an attachment axis; an outer part that is tubular along the attachment axis, and hollow, the inner part being accommodated in the outer part such that the outer surface of the inner part is facing the inner surface of the outer part; an elastomer layer between the outer part and the inner part, the elastomer layer, when seen in section in a longitudinal plane including the attachment axis, including: to either side of the attachment axis, at least three so-called longitudinal portions, and at least two so-called transverse portions.