Question: The Franco-Italian army led by Eugène de Beauharnais was defreated by an Austrian army commanded by a member of what house?

Context paragraphs:
Archduke John of Austria: Archduke John of Austria (German: "Erzherzog Johann Baptist Joseph Fabian Sebastian von Österreich" ; 20 January 1782 – 11 May 1859), a member of the House of Habsburg-Lorraine, was an Austrian field marshal and Imperial regent ("Reichsverweser") of the short-lived German Empire during the Revolutions of 1848.

Piave River 1809 order of battle: The Piave River 1809 Order of Battle shows the units and organization for the Franco-Italian and Austrian Empire armies that fought in the Battle of Piave River on 8 May 1809.  Eugène de Beauharnais, the viceroy of the Kingdom of Italy defeated Archduke John of Austria.  Eugène's Advance Guard crossed the river first and was assailed by Austrian cavalry and artillery.  The French cavalry routed the opposing cavalry and captured 14 enemy guns.  A lull followed as John arranged his infantry in a formidable defensive position.  Meanwhile, Eugène struggled to pour reinforcements into the bridgehead as the Piave rose dangerously.  In the afternoon, the viceroy sent Paul Grenier to drive back the Austrian left while Jacques MacDonald mounted an assault on the center.  The attack succeeded in breaking the Austrian line and compelling John to order a retreat.

Sacile 1809 order of battle: The Battle of Sacile on 16 April 1809 saw the Franco-Italian Army of Italy commanded by Eugène de Beauharnais attack Archduke John of Austria's Army of Inner Austria.  In the morning, the Austrians successfully held off Franco-Italian assaults on their left flank.  Later in the day, John counterattacked to drive Eugène's troops from the battlefield.  The battle at Sacile was preceded by the action of Pordenone on 15 April in which the Austrian advance guard mauled the French rear guard.  The Austrian victory compelled Eugène to withdraw to the Adige River at Verona where he gathered reinforcements and planned a counteroffensive.

Eugénie de Beauharnais: Eugénie Hortense Auguste Napoléone, known as Eugénie de Beauharnais, princess of Leuchtenberg (22 December 1808, Milan – 1 September 1847, Freudenstadt) was a Franco-German princess.  She was the second daughter of Eugène de Beauharnais and Princess Augusta of Bavaria, and a member of the House of Beauharnais.  In 1826 she married Constantine, Prince of Hohenzollern-Hechingen.

Battle of Caldiero (1813): The Battle of Caldiero on 15 November 1813 saw an army of the First French Empire under Eugène de Beauharnais opposed to an Austrian Empire army led by Johann von Hiller.  Eugène, who was the Viceroy of the Napoleonic Kingdom of Italy inflicted a defeat on Hiller's troops, driving them from Caldiero.  The action took place during the War of the Sixth Coalition, part of the Napoleonic Wars.  Caldiero is located 15 km east of Verona on the Autostrada A4.

Battle of Piave River (1809): The Battle of Piave River was fought on 8 May 1809 between the Franco-Italian army under the command of Eugène de Beauharnais and an Austrian army led by Archduke John of Austria.  The Austrian commander made a stand behind the Piave River but he suffered a defeat at the hands of his numerically superior foes.  The combat took place near Nervesa della Battaglia, Italy during the War of the Fifth Coalition, part of the Napoleonic Wars.

Battle of Sacile: The Battle of Sacile (also known as the Battle of Fontana Fredda) on 16 April 1809 and its companion Clash at Pordenone on 15 April saw an Austrian army commanded by Archduke John of Austria defeat a Franco-Italian army led by Eugène de Beauharnais and force it to retreat.  Sacile proved to be the most notable victory of John's career.  The action took place east of the Livenza River near Sacile in modern-day Italy during the War of the Fifth Coalition, part of the Napoleonic Wars.

Battle of Caldiero (1809): In the Battle of Caldiero or Battle of Soave or Battle of Castelcerino from 27 to 30 April 1809, an Austrian army led by Archduke John of Austria defended against a Franco-Italian army headed by Eugène de Beauharnais, the Viceroy of the Kingdom of Italy.  The outnumbered Austrians successfully fended off the attacks of their enemies in actions at San Bonifacio, Soave, and Castelcerino before retreating to the east.  The clash occurred during the War of the Fifth Coalition, part of the Napoleonic Wars.

Battle of Tarvis (1809): The Battle of Tarvis from 16 to 17 May 1809, the Storming of the Malborghetto Blockhouse from 15 to 17 May 1809, and the Storming of the Predil Blockhouse from 15 to 18 May saw the Franco-Italian army of Eugène de Beauharnais attacking Austrian Empire forces under Albert Gyulai.  Eugène crushed Gyulai's division in a pitched battle near Tarvisio, then an Austrian town known as Tarvis.  At nearby Malborghetto Valbruna (Malbotghet Wolfstal) and Predil Pass, small garrisons of Grenz infantry heroically defended two forts before being overwhelmed by sheer numbers.  The Franco-Italian capture of the key mountain passes allowed their forces to invade Austrian Kärnten during the War of the Fifth Coalition.  Tarvisio is located in far northeast Italy, near the borders of both Austria and Slovenia.

Nicolas de Leuchtenberg: Nicolas de Leuchtenberg ("Nicolas Alexander Fritz"; born 12 October 1933, Munich) is a claimant to the Dukedom of Leuchtenberg.  He is the son of Nicolas Nicolaïevitch de Leuchtenberg and his wife Élisabeth Müller-Himmler, and is thus a direct descendent of Alexandre de Beauharnais, Joséphine de Beauharnais, and of Eugène de Beauharnais, first Duke of Leuchtenberg.

Supporting paragraphs: Battle of Sacile Archduke John of Austria
Answer: House of Habsburg-Lorraine