A word for a secret organisation that controls the government

Welcome to the site. I believe your question is appropriate here because it is partly about concept-associative vocabulary.

Four words come to mind, but all (in your 40%) without initial "s", and all with initial "c". They are coterie, cabal, camarilla,and clique. Whether these are used in movies or not I cannot say. The meanings are trivial to look up but of the four I like:

cabal =

a small group of people who plan secretly to take action, especially political action

Cambridge dictionary

and

camarilla =

a group of advisers, usually a secret group who are involved in a plot

for example "... an entourage of penniless, ignorant, intriguing ministers watched over by camarillas of officers..."

Macmillan dictionary

Wikipedia has a broader definition of cabal that fits it almost exactly to your specification:

"A cabal is a group of people united in some close design, usually to promote their private views or interests in an ideology, state, or other community, often by intrigue and usually unbeknownst to those outside their group. The use of this term usually carries negative connotations of political purpose, conspiracy and secrecy."

Wikipedia

An illustration of the use of cabal is:

"The common theme in conspiracy theories about a New World Order is that a secretive power elite with a globalist agenda is conspiring to eventually rule the world through an authoritarian world government—which will replace sovereign nation-states—and an all-encompassing propaganda whose ideology hails the establishment of the New World Order as the culmination of history's progress. Many influential historical and contemporary figures have therefore been alleged to be part of a cabal that operates through many front organizations to orchestrate significant political and financial events, ranging from causing systemic crises to pushing through controversial policies, at both national and international levels, as steps in an ongoing plot to achieve world domination"

Wikipedia

It is interesting that these words reflect a Romance language influence (French & Spanish). If we look for similar words in northern Europe (Scandinavian, German, Russian) we often find transliterative versions of cabal and clique.

I agree Cabal seems the best fit for a general type; an alternative 'S' word of that ilk is:

Sect

Sticking with S, Deep State or Secret State or Shadow State are terms sometimes used for the actual rulers...

A couple of examples of actual names (though not really English usage, but it's possible they have been remembered) -

For specific organisations used in the Bond franchise, beginning with S:

SPECTRE (and before that SMERSH)

For an alleged USA secret society beginning with S (represented among others by George W. Bush):

Skull and Bones

Who are allegedly a branch of the most famous (though not very S-friendly) secret (world) government, The Illuminati.

The first word that came to mind is "cabal". If you plan on using that word, you should keep in mind that it has anti-semitic origins (it comes from the word Kabbalah). Another word is "cartel".

If you're looking for a word that starts with "s", the first word that comes to mind is "syndicate".

Like others, I found that the first term to come to mind was 'cabal'. Bearing in mind your starting-with-S criterion, however, I tried 'secret cabal' -- but although that expression is sometimes used, it seems essentially tautological.

Staying with S, then, and thinking more closely about films, the expression that I suggest is Star Chamber.

A modern Star Chamber would be (or is imagined as!) a hermetic subgroup of any administrative authority, including governments. The concept is essentially conspiratorial: a small group routinely operating without hindrance by its own organisation's published ethical principles and procedures. It is in the nature of a Star Chamber that while some might suspect its existence (perhaps following a surprising coincidence of assassinations or commercial deals), we can never find the evidence to prove it.

The general idea in practice is that, however open and democratic the organisation might be thought to be (and might broadly think itself to be), this small assembly of senior individuals is the group that actually takes all of the important policy, strategic and tactical decisions.

The remaining governance structure is unaware of the Star Chamber, and is there for two main reasons. The legitimate structure of departments and so on provides a superficial public image of fairness, conscience and judiciousness; and it controls the infrastructure that in fact enacts the decisions made by the Star Chamber. Part of the point is that that infrastructure can operate in the sincere belief that its published principles are being upheld.

In conventional imagination, membership of a Star Chamber is achieved only through invitation. If a place becomes available (e.g. through death), or if specific expertise or energy is newly required, then the existing Star Chamber members can identify, and agree to include, a chosen individual.

In turn, that leads to the idea of conspiracies-within-conspiracies. Just possibly, by carefully influencing membership a small number of Star Chamber members might conspire to subvert and control the policies and mission of the Star Chamber itself.

If I am on the mark, you are probably thinking of the 1983 film The Star Chamber, directed by Peter Hyams and starring Michael Douglas. Douglas plays a frustrated young lawyer who wants to take a short cut to simply punishing influential wrongdoers, rather than constantly seeing them go free through a legal system that they routinely manipulate.

The expression 'Star Chamber' itself is certainly much older. Wikipedia suggests several possible origins in English government, perhaps going back as far as Old English, and definitely figuring in later expedient decision-making in Westminster. A summary offered by the National Archive outlines that history, formally starting in the 15th Century.