Where VAT's concerned, nothing is ever simple!
If you join a sports club primarily to participate in the relevant sport then (because your subscription is offered to to you in order for you to access sporting facilities) the subscription is exempt from VAT.
However if a club offers 'social membership' (whereby a subscription allows the member to use the club bar or restaurant but doesn't provide access to sporting facilities) then there is no VAT exemption.
Further, if a club offers 'social membership', it's possible that HMRC might consider that 'full membership' constitutes 'multiple supply', with the part of a 'full' subscription which gives access to sports facilities being exempt from VAT but the 'social' element being subject to VAT.
Further, VAT exemptions only apply when the club is an 'eligible body', which requires it to have a constitution declaring it to be non-profit making and that its income will solely be used for furthering the activities of the club. (So a club which gives part of its profits to local charities risks losing the VAT exemptions which would normally apply)
Notices 701/45
https://www.gov.uk/government/publications/vat-notice-70145-sport/vat-notice-70145-sport
and 701/5 apply
https://www.gov.uk/government/publications/vat-notice-7015-clubs-and-associations/vat-notice-7015-clubs-and-associations#subscriptions