Finite lattice representation problem checking [Grätzer and Schmidt 1963] proves that every algebraic lattice is isomorphic to the congruence lattice of a universal algebra. A finite lattice is algebraic. The finite lattice representation problem asks whether every finite lattice is the congruence lattice of a finite universal algebra. The answer is generally suspected to be no.
Question 1: What is the smallest lattice which is not known to be a congruence lattice of a finite universal algebra?  
Remark: By "smallest" I mean "of smallest cardinal". By "not known" I don't mean that it should have been conjectured somewhere to be a counter-example.   
[Pálfy and Pudlák 1980] proves that the above problem is equivalent to ask whether every finite lattice is the lattice of an interval in the subgroup lattice of a finite group. As pointed out by the authors, this theorem does not imply that every congruence lattice of a finite universal algebra is the lattice of an interval in the subgroup lattice of a finite group (whereas the converse is true), it is just an equivalence between two problems.  
Question 2:  What is the smallest lattice which is not known to be the lattice of an interval in the subgroup lattice of a finite group?
Remark: same as above.
This short course by William DeMeo provides a clear review of the background for this problem.
 A: Answer to Question 1:

All other lattices of size at most 7 are known to be representable as the congruence lattice of a finite algebra.  See my thesis and this MO question.
Answer to Question 2: The same lattice that answers question 1. 
However, there may be other 7-element lattices that are not known to occur as intervals in subgroup lattices of finite groups.
It is know that every lattice of size at most 6 is "group representable" in the above sense. See Watatani (1996) MR1409040 and Aschbacher (2008) MR2393428.
A: If an interval of finite groups $[H,G]$ is lattice-isomorphic to William's lattice then $|G:H|\ge 32$.  
gap> TestDeMeoTransitive(2,30);
[  ]


TransitiveIntermediate:=function(d,r)
    local G,H,a,N,s;
    G:=TransitiveGroup(d,r);
    H:=Stabilizer(G,1);
    N:=IntermediateSubgroups(G,H);
    return N;
end;;

TestDeMeoTransitive:=function(d1,d2)
    local LL,d,n,r,int,sub,inc,L,l,i,j,G;
    LL:=[];
    for d in [d1..d2] do
        n:=NrTransitiveGroups(d);
        for r in [1..n] do
            G:=TransitiveGroup(d,r);
            if IsPrimitive(G) and not IsSolvable(G) then
                int:=TransitiveIntermediate(d,r);
                sub:=int.subgroups; inc:=int.inclusions;
                if Length(sub)=5 and Length(inc)=9 then
                   L:=List(Filtered(inc,i->i[1]=0),j->j[2]);
                   if Length(L) = 3 and Length(Filtered(inc,(i->i[1] in L) and i[2]=6)) = 1 then
                       Add(LL,[d,r]);
                   fi;
                fi;
            fi;
        od;
    od;
    return LL;
end;;

