A normal intermediate subgroup in $B_3$ lattice?

Question

Let $G$ be a finite group and $H$ a subgroup.
Let $\mathcal{L}(H \subset G )$ be the lattice of intermediate subgroups between $H$ and $G$.
An intermediate subgroup $H \subset K \subset G$ is a normal intermediate subgroup if $HgK = KgH$, $\forall g \in G$

Suppose that $\mathcal{L}(H \subset G ) \sim B_3$, as follows:

Let $K_i$ ($i=1,2,3$) be the minimal overgroups of $H$.
Question: Is there $i$ such that $K_i$ is a normal intermediate subgroup of $(H \subset G )$ ?

Remark: I've checked it's true by GAP, for $\vert G \vert <144$, except $64, 96, 128$.

Answer 1

No, the example of @ahulpke here is a counterexample:

First, the program IsNormalIntermediate is the following:

IsNormalIntermediate:=function(G,H,K)
    local D1,D2,s,i,j,E1,E2,c;
    D1:=DoubleCosets(G,H,K);
    s:=Size(DoubleCosets(G,H,K));
    c:=0;
    if s=Size(DoubleCosets(G,K,K)) then
        return true;
    else
        D2:=DoubleCosets(G,K,H);
        for i in [1..s] do 
            for j in [1..s] do
                E1:=Elements(D1[i]);
                E2:=Elements(D2[j]);
                if Size(E1)=Size(E2) then
                    if E1=E2 then
                        c:=c+1;
                    fi;
                fi;
            od;
        od; 
    fi;
    return c=s;
end;;

Next, the couterexample of @ahulpke is:

gap> G:=AlternatingGroup(8);
Alt( [ 1 .. 8 ] )
gap> H:=TransitiveGroup(8,29);
E(8):D_8=[2^3]D(4)
int:=IntermediateSubgroups(G,H);;

because the lattice is equivalent to $B_3$:

 gap> int.inclusions;
[ [ 0, 1 ], [ 0, 2 ], [ 0, 3 ], [ 1, 4 ], [ 1, 6 ], [ 2, 4 ], [ 2, 5 ], [ 3, 5 ], [ 3, 6 ], [ 4, 7 ], [ 5, 7 ], [ 6, 7 ] ]

and the $K_i$ $(i=1,2,3)$ are not normal intermediate:

gap> K1:=int.subgroups[1];;   K2:=int.subgroups[2];;   K3:=int.subgroups[3];;
gap> IsNormalIntermediate(G,H,K1) or IsNormalIntermediate(G,H,K2) or IsNormalIntermediate(G,H,K3);
false

Remark: The others intermediate subgroups are also not normal intermediate:

gap> K4:=int.subgroups[4];;   K5:=int.subgroups[5];;   K6:=int.subgroups[6];;
gap> IsNormalIntermediate(G,H,K4) or IsNormalIntermediate(G,H,K5) or IsNormalIntermediate(G,H,K6);
false