World anatomists got together in Frankfort, Germany, around 1875 to agree on nomenclature. In defining "anatomical position" they decided that a horizontal plane formed by left and right poria and left orbitale would define anatomical position for the head, hence "Frankfort Horizontal."
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Since a hexagon is a horizontal plane figure it cannot have a horizontal plane of symmetry.
vertical - - - - - - - - - - horizontal -------------------------------
A line which divides a body into two parts, such that each part is the mirror image of the others, then the line is called a line of symmetry. If such a line is parallel to the horizontal plane, then it is called a horizontal line of symmetry. Else, if the line of symmetry is perpendicular to the horizontal plane then it is a vertical line of symmetry.
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A vertical or horizontal stabilizer helps the plane to be balanced