This sounds like a relatively recently created body, so there may not be a reliable "official" version in English. Many government sites simply use automatic machine translation.
The two main components of the name (Subdirección General/Promoción Exterior) can each be translated in different ways. I've suggested one version, but others are possible. For example, as the UK is an island nation, everything abroad is "overseas", whereas for other countries "foreign" might be a better option to add to the equation.


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Note added at 22 hrs (2017-04-18 08:04:07 GMT)
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NB: As Robert mentions below, there aren't many government institutions known as Vice directorate in the English-speaking world. However, as I mentioned earlier, many "official" sites simply use automatic machine translation. So, for example in Spain we find things like "The Vice- Directorate of Territorial Coordination " / Subdirección de Arqueología Subacuática (Underwater Archaeology Vice-Directorate) /

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Note added at 23 hrs (2017-04-18 08:05:36 GMT)
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And the Spanish prison system:
"The General Secretariat of Penitentiary Institutions and its dependant General Vice-directorate of Alternative Sentences and Measures, General Vice-directorate of Institutional Relations and Territorial Coordination, General Vice-directorate of Penitentiary Inspection, General Vice-directorate of Penitentiary Management and Treatment, General Vice-directorate of Health Coordination, as well as General Vice-directorate of Human Resources and General Vice-directorate of Services, have their headquarters in Calle Alcalá, nº 38, Madrid
All the General Vice-directorates are coordinated with one another and with every Prison Director and Vice-director; ..."
http://www.institucionpenitenciaria.es/web/portal/idioma/en/...

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Note added at 23 hrs (2017-04-18 08:08:37 GMT)
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Robert suggests a more anglo-friendly version for the English-speaking world. If you decide to go this route there are indeed many options that we could use, although personally I usually prefer to stick to calques or literal translations for the names of institutions, to ensure maximum comprehension, even perhaps at the cost of seeming slightly awkward or old-fashioned to some English-speaking readers.