Question: When was the battalion that deployed to the European Theatrer, landing at Omaha Beach able to establish a beachhead as part of Operation Overlord after a successful "D-Day,"?

Context paragraphs:
Invasion of Normandy: The Western Allies of World War II launched the largest amphibious invasion in history when they assaulted Normandy, located on the northern coast of France, on 6 June 1944.  The invaders were able to establish a beachhead as part of Operation Overlord after a successful "D-Day," the first day of the invasion.

Charles T. Horner Jr.: Charles Thompson Horner Jr. (May 6, 1916 - April 4, 1992) was a highly decorated officer of the United States Army who served in the 16th Infantry Regiment, U.S. 1st Infantry Division from July 1940 to February 1946 and held every position from Platoon Leader to Regimental Commander.  As a leader in the "Big Red One", he participated in the North Africa, Sicily, and the European Campaigns and three amphibious landings; the invasions of North Africa (Operation Torch), Sicily (Operation Husky), and Europe (Operation Overlord) where he commanded the 3rd Battalion, 16th Infantry on Omaha Beach.

12th Engineer Battalion (United States): The 12th Engineer Battalion was activated 1 July 1940, at Ft. Jackson, South Carolina as an organic element of the 8th Infantry Division In July 1944 the battalion deployed to the European Theatre, landing at Omaha Beach in support of the greatest landing operation in history.  Subsequently the Battalion participated in campaigns in Normandy, Northern France, Rhineland, and Central Europe, and Luxembourg

List of ships in Omaha Bombardment Group: Below is a list of ships responsible for bombarding targets at Omaha Beach as part of the Normandy landings on June 6, 1944, the opening day of Operation Overlord, the Allied operation that launched the successful invasion of German-occupied western Europe during World War II.

Heinrich Severloh: Heinrich Severloh (23 June 1923 – 14 January 2006) was a soldier in the German 352nd Infantry Division stationed in Normandy in 1944.  He became infamous for a memoir" WN 62 – Erinnerungen an Omaha Beach Normandie, 6.  Juni 1944," published in 2000.  This book is noted for the authors’ claim that as a machine gunner, Severloh inflicted over 2,000 casualties to American soldiers landing on Omaha Beach on D-Day, Subsequent to the book being published, some have referred to Severloh as “The Beast of Omaha.”  However, Severloh’s claim is not viewed as credible by either US or German historians.  Total US casualties (Killed, Wounded, Missing) from all sources at Omaha Beach on D-Day is estimated at 3,686.

Battle of Port-en-Bessin: The Battle of Port-en-Bessin also known as Operation Aubery took place from 7–8 June 1944, at a small fishing harbour west of Arromanches during the Normandy landings of World War II.  The village was between Omaha Beach to the west in the U.S. V Corps sector, and Gold Beach to the east in the British XXX Corps sector.  An objective during Operation Overlord, the fortified port was captured by No. 47 (Royal Marine) Commando of the 4th Special Service Brigade.

Leonard Lomell: Leonard G. “Bud” Lomell (January 22, 1920 – March 1, 2011) was a highly decorated former United States Army Ranger who served in World War II.  He is best known for his actions in the first hours of D-Day at Pointe du Hoc on the coast of Normandy, France.  Pointe du Hoc was the site of the German Army’s largest coastal weapons, five 155-millimeter German guns with a 25-kilometer range that endangered the tens of thousands of troops landing on Omaha Beach and Utah Beach, and thousands of watercraft in the English Channel supporting the Normandy invasion.  Unbeknownst to the Allied intelligence, the Germans had concealed the guns in an orchard, but left them operational and ready to fire.  Through skill, courage and “pure luck,” Lomell found and quickly disabled all five guns.  Lomell was recognized by historian Stephen Ambrose as the single individual — other than Gen. Dwight D. Eisenhower — most responsible for the success of D-Day.  Six months later, in the Battle of Hürtgen Forest, he would again distinguish himself, earning a Silver Star for his heroism and leadership as the 2nd Ranger Battalion captured and held Hill 400.  After the war he returned to Ocean County, New Jersey, becoming an attorney in Toms River.

10th Battalion (Australia): The 10th Battalion was an infantry battalion of the Australian Army that served as part of the Australian Imperial Force during World War I. Among the first units raised in Australia during the war, the battalion was recruited from South Australia in August 1914 and along with the 9th, 11th and 12th Battalions, it formed part of the 3rd Brigade, 1st Division.  After its personnel completed basic training, the battalion embarked for the Middle East, where further training was undertaken in Egypt before the battalion was committed to the Gallipoli Campaign.  On 25 April 1915, the 10th Battalion took part in the Landing at Anzac Cove, coming ashore during the initial stages of the operation as part of the covering force.  Members from the 10th Battalion penetrated the furthest inland of any Australian troops during the initial fighting, before the Allied advance inland was checked.  After this, the battalion helped defend the beachhead against a Turkish counter-attack in May, before joining the August Offensive, a failed Allied effort to break the deadlock.  Casualties were heavy throughout the campaign and in November 1915, the surviving members were withdrawn from the peninsula and in early 1916 the battalion was reformed in Egypt.  After a period of reorganisation, during which the 10th provided a cadre staff to the newly formed 50th Battalion, it was transferred to the Western Front in March 1916, and for the next two-and-a-half years took part in trench warfare in France and Belgium until the Armistice in 1918.  The last detachment of men from the 10th Battalion returned to Australia in September 1919.

Nicholas Lamia: Nicholas Lamia’s outfit from the Naval Combat Demolition Unit’s (NCDU) Assault Force 'O' of the Western Naval Task Force assaulted Normandy Beach on 6 June 1944.  It had 16 Landing Craft’s Mechanized (LCM’s) that were the initial assault force and was the first force to reach Omaha beach on D-Day.  Nicholas Lamia’s LCM was full of engineers and demolitions troops to clear the beach of obstacles (20 Army 20 Navy).  His job during D-day was to land the men on the beach during the amphibious assaults while he manned the 50 caliber machine gun.  Lamia’s LCM approached the beach under terrible automatic fire and almost without exception every LCM was subjected to mortar, machine gun and German "88" barrages.  As Lamia (with the initial assault force) approached the beach they received fire from a two story building on the hill.  Nicholas Lamia and Joey Kuntz both focused their 50 caliber machine gun fire on the building’s window.  They saw an explosion in the building and the firing from the building ceased.  The boat was cleared of the Engineers under vicious fire and the ramp was lifted.  The moment Lamia’s ramp reached the top closed position, the LCM was hit by a German 88 mm round and tore the ramp open like a tin can, but it did not sink.  Of the 16 LCM’s in NCDU, 13 were sunk or out of commission.

741st Tank Battalion (United States): The 741st Tank Battalion was an independent tank battalion that participated in the European Theater of Operations with the United States Army in World War II.  The battalion participated in combat operations throughout northern Europe until V-E Day.  It was one of five tank battalions (all independent) that landed in Normandy on D-Day (6 June 1944).  It landed on Omaha Beach supporting the 1st Infantry Division, but was attached to 2d Infantry Division on 15 June 1944, which it supported for most of the reminder of the war.  The battalion played a key role in blunting the northern flank of the German attack during the Battle of the Bulge in December 1944.  The 741st Tank Battalion advanced as far as Plzeň, Czechoslovakia by the end of the war.

Supporting paragraphs: 12th Engineer Battalion (United States) Invasion of Normandy
Answer: 6 June 1944