Not a Lattice. For a Poset to be Lattice, For every pair of elements, LUB and GLB both must exist. "In any Poset, for any subset of elements, LUB and GLB are always Unique, if Exists." For the given Hasse Diagram, Some of the Properties are as follows:
$UB\left \{b,c \right \} = \left \{ g,f,h \right \} // \,NO \,\,\,LUB$
$LB\left \{ b,c \right \} = \left \{ a\right \} // Also \,\,GLB$
$UB\left \{ b,e \right \} = \left \{ g,f,h\right \} // NO \,\,LUB$
$LB\left \{ b,e \right \} = \left \{ a\right \} // Also \,\,GLB$
$UB\left \{ d,c \right \} = \left \{ f,h\right \} // \,\,LUB = f$
$LB\left \{ d,c \right \} = \left \{ a\right \} // Also \,\,GLB$
$UB\left \{ f,g \right \} = \left \{ h\right \} // Also \,\,LUB$
$LB\left \{ f,g \right \} = \left \{ b,e,c,a\right \} // NO \,\,GLB$