Biblatex move organization

Question

I would like the institution to be shown between author and title. How do I have to change my code in order to achieve that?

\begin{filecontents}{\jobname.bib}
        @online{musk,
            author       = {Elon Musk},
            title        = {Tesla Model X},
            url          = {https://www.tesla.com/modelx?redirect=no},
            date         = {2017-07-04},
            organization = {Tesla},
            urldate      = {2018-10-16},
        }
    \end{filecontents}



\documentclass{article}
\usepackage[style=authortitle,]{biblatex}
\addbibresource{\jobname.bib}
\DeclareListFormat{organization}{%
    \usebibmacro{list:delim}{#1}%
    \mkbibemph{#1}\isdot
    \usebibmacro{list:andothers}}


\begin{document}
    Let's cite! \footcite{musk}
    \printbibliography
\end{document}

Answer 1

You can redefine online's bibdriver to that purpose:

\begin{filecontents}{\jobname.bib}
  @online{musk,
    author       = {Elon Musk},
    title        = {Tesla Model X},
    url          = {https://www.tesla.com/modelx?redirect=no},
    date         = {2017-07-04},
    organization = {Tesla},
    urldate      = {2018-10-16},
  }
\end{filecontents}

\documentclass{article}
\usepackage[style=authortitle]{biblatex}
\addbibresource{\jobname.bib}
\DeclareListFormat{organization}{%
  \usebibmacro{list:delim}{#1}%
  \mkbibemph{#1}\isdot
  \usebibmacro{list:andothers}}

\DeclareBibliographyDriver{online}{%
  \usebibmacro{bibindex}%
  \usebibmacro{begentry}%
  \usebibmacro{author/editor+others/translator+others}%
  \setunit{\printdelim{nametitledelim}}\newblock
  \printlist{organization}%
  \newunit\newblock
  \usebibmacro{title}%
  \newunit
  \printlist{language}%
  \newunit\newblock
  \usebibmacro{byauthor}%
  \newunit\newblock
  \usebibmacro{byeditor+others}%
  \newunit\newblock
  \printfield{version}%
  \newunit
  \printfield{note}%
  \newunit\newblock
  \usebibmacro{date}%
  \newunit\newblock
  \iftoggle{bbx:eprint}
    {\usebibmacro{eprint}}
    {}%
  \newunit\newblock
  \usebibmacro{url+urldate}%
  \newunit\newblock
  \usebibmacro{addendum+pubstate}%
  \setunit{\bibpagerefpunct}\newblock
  \usebibmacro{pageref}%
  \newunit\newblock
  \iftoggle{bbx:related}
    {\usebibmacro{related:init}%
     \usebibmacro{related}}
    {}%
  \usebibmacro{finentry}}

\begin{document}
Let's cite! \footcite{musk}
\printbibliography
\end{document}