Regular {3, 2n} maps

Weddslist has a listing of regular maps.

A necessary condition for a regular map to be a CFTM is that its Schläfli formula be {3, 2n} for some positive integer n.

Regular maps meeting this condition include:

Orientable genus 0
di-triangle {3, 2} (it seems the only configuration is folded)
octahedron {3,4}

Orientable genus 1
{3, 6} (1, 1)  (only one vertex)
{3, 6} (0, 2)  6 triangles ()(()); ((())); (())() (it seems the only configuration is folded)
{3, 6} (2, 2)  8 triangles
{3, 6} (1, 3) 14 triangles
{3, 6} (3, 3) 18 triangles
{3, 6} (0, 4) 24 triangles
{3, 6} (2, 4) 26 triangles
{3, 6} (4, 4) 32 triangles
etc.

Orientable genus 2
S2:{3, 8} 16 triangles

Non-orientable genus 1
hemioctahedron {3, 4} 4 triangles

Non-orientable genus 6
C6: {3, 10} 10 20 triangles
C6: {3, 10} 5 20 triangles