openssl x509: If you mean the Subject and/or Issuer field(s), the simplest and most readable way (IMO) is
openss x509 -in certpemfile -noout -text -nameopt multiline,show_type or if you want only the name field(s) change -text to -subject and/or -issuer. There are other formats, and if you want non-trivially encoded data to display usably you may need additional options; see the man page on your system or at https://www.openssl.org/docs/apps/x509.html#NAME-OPTIONS . If your input is "DER" (binary) not the much more common "PEM" (textual), add -inform der.
If you mean extensions than can contain (Distinguished)Name fields -- I count SAN/IAN, NameConstraints, CRLDP, AIA/SIA, although I don't see how DN would be useful in the latter three, and there are surely more -- that's not supported that I know of. But see next.
other: openssl asn1parse dumps the exact types, lengths, and mostly values of any ASN.1 BER/DER structure, including a cert. For X.509-form extensions (wrapped in OCTET STRING) you can use -strparse offset to ASN.1-decode the nested structure. Identifying and selecting desired fields is your job and in general can be difficult, but since both DNs and extensions label each of their elements with a preceding OID (which OpenSSL calls just OBJECT) and asn1parse decodes all OIDs known to openssl (and you can give it additional ones) you can usually just visually scan for countryName, commonName and the like. Note on PEM: many openssl functions create or accept and ignore "comments" in PEM-format data, but asn1parse does NOT. It will handle the dash-BEGIN and dash-END lines, but almost anything else it will treat as part of the data causing wrong results. Verify your input file (or pipe or whatever) contains only the one cert.