Decimal alignment in a table with math mode

You are probably looking for \multicolumn:

\documentclass{article}
\usepackage[utf8]{inputenc}
\usepackage{booktabs}
\usepackage{multirow}
\usepackage{ctable}
\usepackage{tabularx}
\usepackage{graphicx}
\usepackage{dcolumn}
\newcolumntype{d}[1]{D{.}{.}{#1} }


\begin{document}

\begin{table}[]
  \centering
  \begin{tabular}{@{}lc*{4}{d{1.5}}@{}}
    \toprule
    {} & {} & \multicolumn{2}{c}{Group A} & \multicolumn{2}{c}{Group B} \\
    \cmidrule(lr){3-4}  \cmidrule(l){5-6}
    Types & Character & \multicolumn{1}{c}{Results A} & \multicolumn{1}{c}{Results B} & \multicolumn{1}{c}{Results A} & \multicolumn{1}{c}{Results B}  \\
    \midrule
    \multirow{3}{*}{Main}
    {}    & The Good & .111      & .789_{b}^{b}  & .520      & .555_{b}^{a}  \\
    {}    & The Bad  & .111      & .636_{b}^{al} & .520      & .730_{b}^{al} \\
    {}    & The Ugly & .111      & .525_{b}      & .520      & .696_{b}      \\
    \bottomrule
  \end{tabular}
  \caption{Caption}
  \label{tab:my_label}
\end{table}

\end{document}

However, as mentioned in the comments, I think you should reconsider the case against siunitx. More important than being a "basic package" is the fact that it is very well crafted and zealously maintained, besides being widely used. (This is not a judgement on dcolumn of which I'm not an user). You should, of course, decide that along with your work group.

The same table done with siunitx would be something like:

\documentclass{article}

\usepackage{booktabs}
\usepackage{multirow}
\usepackage{siunitx}

\begin{document}

\begin{table}
  \centering
  \sisetup{table-format = 0.3}
  \begin{tabular}{@{}
    lc
    S
    S[table-space-text-post = {$_{b}^{al}$}] % make it the largest post-number element
    S
    S[table-space-text-post = {$_{b}^{al}$}]
    @{}}
    \toprule
          &           & \multicolumn{2}{c}{Group A}     & \multicolumn{2}{c}{Group B}    \\
    \cmidrule(lr){3-4}  \cmidrule(l){5-6}
    Types & Character & {Results A} & {Results B}       & {Results A} & {Results B}      \\
    \midrule
    \multirow{3}{*}{Main}
          & The Good  & .111        & .789{$_{b}^{b}$}  & .520        & .555{$_{b}^{a}$}  \\
          & The Bad   & .111        & .636{$_{b}^{al}$} & .520        & .730{$_{b}^{al}$} \\
          & The Ugly  & .111        & .525{$_{b}$}      & .520        & .696{$_{b}$}      \\
    \bottomrule
  \end{tabular}
  \caption{Caption}
  \label{tab:my_label}
\end{table}

\end{document}

It does automatically take care of the leading zero, emphasized by @Denis in the comments.

You can (should, in my opinion) use siunitx; here I show also how to have real note markers upright that leave the exact space for them.

If you want to set some of the entries bold, there is a trick adapted from https://tex.stackexchange.com/a/352028/4427

\documentclass{article}
\usepackage[utf8]{inputenc}
\usepackage{booktabs}
\usepackage{siunitx}

\NewDocumentCommand{\bs}{}{\fontseries{b}\selectfont}

\makeatletter
\NewDocumentCommand{\tss}{mm}{%
  {\m@th\ensuremath{%
    ^{\mbox{\fontsize\sf@size\z@\selectfont #1}}%
    _{\mbox{\fontsize\sf@size\z@\selectfont #2}}%
  }}%
}
\makeatother

\begin{document}

\begin{table}[htp]
\centering

\sisetup{detect-weight,mode=text}
\DeclareDocumentCommand{\bfseries}{}{\bs}

\begin{tabular}{
  @{}
  l
  c
  S[table-format=1.3]
  S[table-format=1.3,table-space-text-post=\tss{al}{b}]
  S[table-format=1.3]
  S[table-format=1.3,table-space-text-post=\tss{al}{b}]
  @{}
}
\toprule
 & & \multicolumn{2}{c}{Group A} & \multicolumn{2}{c}{Group B} \\
\cmidrule(lr){3-4}  \cmidrule(l){5-6}
Types & Character & {Results A} & {Results B} & {Results A} & {Results B} \\
\midrule
Main & The Good & \bs .111 & .789\tss{b}{b}  & .520 & .555\tss{a}{b}  \\
     & The Bad  &     .111 & .636\tss{al}{b} & .520 & .730\tss{al}{b} \\
     & The Ugly &     .111 & .525\tss{}{b}   & .520 & .696\tss{}{b}   \\
\bottomrule
\end{tabular}

\caption{Caption}
\label{tab:my_label}

\end{table}

\end{document}

I wouldn't use \multirow: blank space below “Main” means repetition. Shifting it down makes quite unclear whether “Main” refers to all rows or just the middle one.